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182 The Prime Number Theorem The method we used to derive Theorem 6.9 is very ﬂexible, and can be applied to many other situations. For example, the summatory function M (x ) = ∑ n≤x µ(n) can be estimated by applying the above method with ζ′/ζ replaced by 1 /ζ. Thus it may be shown that M (x ) ≪ x exp ( − c √ log x ) (6.17) for x ≥ 2. If instead we were to apply the method to the function 1 /ζ(s + 1), we would ﬁnd that ∑ n≤x µ(n) n ≪ exp ( − c √ log x ) , (6.18) since 1 /(sζ(s + 1)) is analytic at s = 0. Hence in particular, ∞∑ n=1 µ(n) n = 0. (6.19) 6.2.1 Exercises 1. (Landau 1901b; cf. Rosser & Schoenfeld 1962) Use Theorem 6.9 to show that π(2x ) − 2π(x ) =− 2(log 2) x (log x )−2 + O (x (log x )−3 ). Deduce that for all large x , the interval ( x ,2x ] contains fewer prime num- bers than the interval (0 ,x ]. 2. Use Theorem 6.9 to show that if n is of the form n = ∏ p≤y p where y is sufﬁciently large, then d (n) > n(log 2) /log log n . 3. (a) Use Theorem 6.9 to show that ∑ x <p≤y 1 p = log log y log x + O ( exp ( − c √ log x )) . (b) Use the above and Theorem 2.7 to show that ∑ p≤x 1 p = log log x + b + O ( exp ( − c √ log x )) where b = C0 − ∑ p ∑ ∞ k=2 1/(kp k ). 4. Show that for x ≥ 2, ∑ n≤x /Lambda1 (n) n = log x − C0 + O ( exp ( − c √ log x )) .	{ "creation_date": "2025-02-03", "file_name": "(Cambridge Studies in Advanced Mathematics 97) Hugh L. Montgomery, Robert C. Vaughan - Multiplicative number theory I_ Classical theory-Cambridge University Press (2006).pdf", "file_path": "/home/suryaremanan/Documents/IMO/augmentoolkit/original/input/(Cambridge Studies in Advanced Mathematics 97) Hugh L. Montgomery, Robert C. Vaughan - Multiplicative number theory I_ Classical theory-Cambridge University Press (2006).pdf", "file_size": 2365282, "file_type": "application/pdf", "last_modified_date": "2025-02-03", "page_label": "182" }
6.2 The Prime Number Theorem 183 5. (cf. Cipolla 1902; Rosser 1939) Let p1 < p2 < ··· denote the prime num- bers. Show that pn = n ( log n + log log n − 1 + log log n log n − 2 log n + O ((log log n)2 (log n)2 ) . 6. (Landau 1900) Let πk (x ) denote the number of integers not exceeding x that are composed of exactly k distinct primes. (a) Show that π2 (x ) = ∑ p≤√x π(x /p) + O ( x (log x )−2 ) . (b) Show that the sum above is ∑ p≤√x x p log x /p + O ( x (log log x )(log x )−2 ) . (c) Using Theorem 6.9 and integration by parts, show that the sum above is x ∫ √x 2 du u(log x /u) log u + O (x /log x ). (d) Conclude that π2 (x ) = x (log log x )/log x + O (x /log x ). 7. (D. E. Knutson) Let dn denote the least common multiple of the numbers 1,2,..., n. (a) Show that dn = exp(ψ(n)). (b) Let E (z) = ∑ ∞ n=1 zn /dn . Show that this power series has radius of convergence e. (c) Show that E (1) is irrational. 8. (Landau 1905) Let Q(x ) denote the number of square-free integers not exceeding x , and deﬁne R(x ) by the relation Q(x ) = (6/π2 )x + R(x ). (a) Show that R(x ) = M ( y){x /y2 }− ∑ d ≤y µ(d ){x /d 2 } + ∑ m≤x /y2 M (√ x /m ) − 2x ∫ ∞ y M (u)u−3 du . (b) T aking y = x 1/2 exp(−c√ log x ) where c is sufﬁciently small, show that R(x ) ≪ x 1/2 exp(−c√ log x ). 9. Let N = N ( Q) = 1 + ∑ q ≤Q ϕ(q ) be the number of Farey points of order Q, and for 0 ≤ α ≤ 1 write card{(a,q ): q ≤ Q, (a,q ) = 1, a/q ≤ α}= N α + R	{ "creation_date": "2025-02-03", "file_name": "(Cambridge Studies in Advanced Mathematics 97) Hugh L. Montgomery, Robert C. Vaughan - Multiplicative number theory I_ Classical theory-Cambridge University Press (2006).pdf", "file_path": "/home/suryaremanan/Documents/IMO/augmentoolkit/original/input/(Cambridge Studies in Advanced Mathematics 97) Hugh L. Montgomery, Robert C. Vaughan - Multiplicative number theory I_ Classical theory-Cambridge University Press (2006).pdf", "file_size": 2365282, "file_type": "application/pdf", "last_modified_date": "2025-02-03", "page_label": "183" }
184 The Prime Number Theorem where R = R( Q,α). (a) Show that if α = (1/Q)−, then R =− N /Q ≍− Q. (b) Show that if α = 1 − 1/Q, then R = N /Q − 1 ≍ Q. (c) Show that R =− ∑ r ≤Q {r α}M ( Q/r ) for 0 ≤ α ≤ 1. (d) Show that R ≪ Q uniformly for 0 ≤ α ≤ 1. 10. (Landau 1903b; Massias, Nicolas & Robin 1988, 1989) Let f (n) denote the maximal order of any element of the symmetric group Sn . (a) Show that f (n) = max lcm( n1 ,n2 ,..., nk ) where the maximum is ex- tended over all sets {n1 ,n2 ,..., nk ) of natural numbers for which n1 + n2 +···+ nk ≤ n. (b) Choose y as large as possible so that ∑ p≤y p ≤ n. Show that log f (n) ≥ ∑ p≤y log p = (1 + o(1))(n log n)1/2 . (c) Show that f (n) = max q1 q2 ··· qk where qi = pa(i ) i , pi ̸=p j for i ̸= j , and ∑ qi ≤ n. (d) Use the arithmetic–geometric mean inequality to show that ∏ qi ≤ (n/k)k . (e) Show that if k is the number of qi ’s in (c), then k ≤ (2 + o(1))(n/log n)1/2 . (f) Conclude that log f (n) ≍ (n log n)1/2 . 11. Let λ(n) = (−1)/Omega1 (n) be Liouville’s lambda function. (a) Show that ∑ ∞ n=1 λ(n)n−s = ζ(2s)/ζ(s) for σ> 1. (b) Using the method of proof of Theorem 6.9, show that ∑ n≤x λ(n) ≪ x exp ( − c √ log x ) . (c) Use (6.17) and the fact that λ(n) = ∑ d 2 \|n µ(n/d 2 ) to give a second proof of the above estimate. 12. (Landau 1907, Section 14) Let cn = 1i f n is a prime or a prime power, cn = 0 otherwise. (a) Show that µ(n)ω(n) =− ∑ d \|n cd µ(n/d ). (b) Use (6.18) and the method of the hyperbola to show that ∞∑ n=1 µ(n)ω(n) n = 0.	{ "creation_date": "2025-02-03", "file_name": "(Cambridge Studies in Advanced Mathematics 97) Hugh L. Montgomery, Robert C. Vaughan - Multiplicative number theory I_ Classical theory-Cambridge University Press (2006).pdf", "file_path": "/home/suryaremanan/Documents/IMO/augmentoolkit/original/input/(Cambridge Studies in Advanced Mathematics 97) Hugh L. Montgomery, Robert C. Vaughan - Multiplicative number theory I_ Classical theory-Cambridge University Press (2006).pdf", "file_size": 2365282, "file_type": "application/pdf", "last_modified_date": "2025-02-03", "page_label": "184" }
6.2 The Prime Number Theorem 185 13. Use the method of proof of Theorem 6.9 to show that ∑ n≤x /Lambda1 (n)n−it = x 1−it 1 − it + O (x exp ( − c √ log x ) + O ( x (log x )2 exp ( −c log x log τ )) uniformly for \|t \|≤ x . 14. Use the method of proof of Theorem 6.9 to show that for any ﬁxed real t , ∞∑ n=1 µ(n)n−1−it = 1 ζ(1 + it ) . 15. (a) Use the method of proof of Theorem 6.9 to show that for any ﬁxed t ̸=0, ∞∑ n=1 /Lambda1 (n) log n n−1−it = log ζ(1 + it ). (b) Deduce that for any t ̸=0, ∏ p (1 − p−1−it )−1 = ζ(1 + it ). 16. (Landau 1899b, 1901a, 1903c) Use the method of proof of Theorem 6.9 to show that (a) ∞∑ n=1 µ(n) log n n =− 1; (b) ∞∑ n=1 µ(n)(log n)2 n =− 2C0 ; (c) ∞∑ n=1 λ(n) log n n =− ζ(2). 17. T aking (6.18) and a quantitative form of the ﬁrst part of the preceding exercise for granted, use elementary reasoning to show that if q ≤ x then (a) ∑ n≤x (n,q )=1 µ(n) n ≪ exp ( − c √ log x ) , (b) ∑ n≤x (n,q )=1 µ(n) log n n =− q ϕ(q ) + O ( exp ( − c √ log x )) . 18. (Hardy 1921) Use the method of proof of Theorem 6.9 to show that (a) ∞∑ n=1 µ(n) ϕ(n) = 0; (b) ∞∑ n=1 µ(n) log n ϕ(n) = 0;	{ "creation_date": "2025-02-03", "file_name": "(Cambridge Studies in Advanced Mathematics 97) Hugh L. Montgomery, Robert C. Vaughan - Multiplicative number theory I_ Classical theory-Cambridge University Press (2006).pdf", "file_path": "/home/suryaremanan/Documents/IMO/augmentoolkit/original/input/(Cambridge Studies in Advanced Mathematics 97) Hugh L. Montgomery, Robert C. Vaughan - Multiplicative number theory I_ Classical theory-Cambridge University Press (2006).pdf", "file_size": 2365282, "file_type": "application/pdf", "last_modified_date": "2025-02-03", "page_label": "185" }
186 The Prime Number Theorem (c) ∞∑ n=1 µ(n)(log n)2 ϕ(n) = 4 A log 2 where A = ∏ p>2 ( 1 − 1 ( p−1)2 ) . 19. Let Q(x ) denote the number of square-free integers not exceeding x , and recall Theorem 2.2. (a) Show that Q(x ) = 6 π2 x − x ∑ n>√x µ(n) n2 − ∑ n≤√x µ(n){x /n2 } where {θ}= x − [x ] is the fractional part of θ. (b) Show that ∑ n>y µ(n)/n2 ≪ y−1 exp(−c√ log y) for y ≥ 2. (c) Note that if k is a positive integer, then {x /n2 } is monotonic for n in the interval √x /(k + 1) < n ≤ √x /k. Deduce that if x ≥ 2k2 , then ∑ √x /(k+1)<n≤√x /k µ(n){x /n2 }≪ √ x /k exp ( − c √ log x ) . (d) By using the above for 1 ≤ k ≤ K = exp(−b√ log x ) where b is suit- ably chosen in terms of c, show that Q(x ) = 6 π2 x + O ( x 1/2 exp ( − c 2 √ log x )) . 20. (Ingham 1945) Let F (n) = ∑ d \|n f (d ) for all n. From our remarks at the beginning of Chapter 2 we see that it is natural to expect a connection between (i) S(x ): = ∑ n≤x F (n) = cx + o(x ); (ii) ∑ ∞ n=1 f (n)/n = c. Neither of these implies the other, but we show now that (i) implies that the series (ii) is (C,1) summable toc. (a) Show that S(x ) = ∑ n≤x f (n)[x /n]. (b) Show that ∑ n≤x f (n) n ( 1 − n x ) = ∫ x 1 S(v) (∑ d ≤x /v µ(d )/d ) d v v2 . (c) Show that ∫ x 1 ∑ d ≤x /v µ(d ) d d v v → 1 as x →∞ .	{ "creation_date": "2025-02-03", "file_name": "(Cambridge Studies in Advanced Mathematics 97) Hugh L. Montgomery, Robert C. Vaughan - Multiplicative number theory I_ Classical theory-Cambridge University Press (2006).pdf", "file_path": "/home/suryaremanan/Documents/IMO/augmentoolkit/original/input/(Cambridge Studies in Advanced Mathematics 97) Hugh L. Montgomery, Robert C. Vaughan - Multiplicative number theory I_ Classical theory-Cambridge University Press (2006).pdf", "file_size": 2365282, "file_type": "application/pdf", "last_modified_date": "2025-02-03", "page_label": "186" }
6.2 The Prime Number Theorem 187 (d) Use the estimate ∑ d ≤y µ(d )/d ≪ (log 2 y)−2 to show that ∫ x 1 ⏐ ⏐ ⏐ ⏐ ⏐ ∑ d ≤x /v µ(d ) d ⏐ ⏐ ⏐ ⏐ ⏐ d v v ≪ 1. (e) Mimic the proof of Theorem 5.5, or use Exercise 5.2.6 to show that if (i) holds, then lim x →∞ ∑ n≤x f (n) n ( 1 − n x ) = c. (f) Use Theorem 5.6 to show that if (i) holds and f (n) = O (1), then (ii) follows. (g) T ake f (n) = µ(n) to deduce that ∑ ∞ n=1 µ(n)/n = 0. (Of course we used much more above in (d). For a result in the converse direction, see Exercise 8.1.5.) 21. (Landau 1908b) Let R be the set of positive integers that can be expressed as a sum of two squares, let R(x ) denote the number of such integers not exceeding x , and let χ1 denote the non-principal character (mod 4), as in Exercise 6.1.5. (a) Show that ∑ n∈R n−s = (1 − 2−s )−1 ∏ p≡1 (4) (1 − p−s )−1 ∏ p≡3 (4) (1 − p−2s )−1 for σ> 1. (b) Show that the Dirichlet series above is f (s)√ ζ(s)L (s,χ1 ) where f (s) = (1 − 2−s )−1/2 ∏ p≡3 (4) (1 − p−2s )−1/2 is a Dirichlet series with abscissa of convergence σc = 1/2. (c) Deduce that the Dirichlet series generating function for R has a quadratic singularity at s = 1. (d) Show that R(x ) = 1 2πi ∫ C f (s) √ ζ(s)L (s,χ1 ) x s s ds + O ( x exp ( − c √ log x )) where C is the contour running from 1 − c − i δ along a straight line to 1 − i δ, then along the semicircle 1 + δei θ, −π/2 ≤ θ ≤ π/2, and ﬁnally along a straight line to 1 − c + i δ. Here c should be sufﬁciently small and δ = 1/log x . (e) Show that the integral above is = 1 2πi ∫ C g(s)x s √s − 1 ds	{ "creation_date": "2025-02-03", "file_name": "(Cambridge Studies in Advanced Mathematics 97) Hugh L. Montgomery, Robert C. Vaughan - Multiplicative number theory I_ Classical theory-Cambridge University Press (2006).pdf", "file_path": "/home/suryaremanan/Documents/IMO/augmentoolkit/original/input/(Cambridge Studies in Advanced Mathematics 97) Hugh L. Montgomery, Robert C. Vaughan - Multiplicative number theory I_ Classical theory-Cambridge University Press (2006).pdf", "file_size": 2365282, "file_type": "application/pdf", "last_modified_date": "2025-02-03", "page_label": "187" }
188 The Prime Number Theorem where g(s) = f (s) s √ (s − 1)ζ(s)L (s,χ1 ) is analytic in a neighbourhood of 1. (f) Show that g(1) = √ π 2 ∏ p≡3 (4) (1 − p−2 )−1/2 . (g) Show that g(s) = g(1) + O (\|s − 1\|) when s is near 1. (h) By means of Theorem C.3 with s = 1/2, or otherwise, show that 1 2πi ∫ C x s √s − 1 ds = x√π log x + O (x 1−c ). (i) Show that if δ = 1/log x , then ∫ C \|s − 1\|1/2 x σ \|ds \|≪ x (log x )3/2 . (j) Show that R(x ) = bx√ log x + O ( x (log x )−3/2 ) where b = 2−1/2 ∏ p≡3 (4) (1 − p−2 )−1/2 . 22. Let A denote the set of those positive integers that are composed entirely of the prime 2 and primes ≡ 1 (mod 4), and let B be the the set of those positive integers that are composed entirely of primes ≡ 3 (mod 4). (a) Explain why any positive integer n has a unique representation in the form n = a(n)b(n) where a(n) ∈ A and b(n) ∈ B. (b) Let A(x ) denote the number of a ∈ A, a ≤ x . Show that A(x ) = αx√ log x + O ( x (log x )3/2 ) where α = 1/ √ 2. (c) Let B (x ) denote the number of b ∈ B, b ≤ x . Show that B (x ) = βx√ log x + O ( x (log x )3/2 ) where β = √ 2/π.	{ "creation_date": "2025-02-03", "file_name": "(Cambridge Studies in Advanced Mathematics 97) Hugh L. Montgomery, Robert C. Vaughan - Multiplicative number theory I_ Classical theory-Cambridge University Press (2006).pdf", "file_path": "/home/suryaremanan/Documents/IMO/augmentoolkit/original/input/(Cambridge Studies in Advanced Mathematics 97) Hugh L. Montgomery, Robert C. Vaughan - Multiplicative number theory I_ Classical theory-Cambridge University Press (2006).pdf", "file_size": 2365282, "file_type": "application/pdf", "last_modified_date": "2025-02-03", "page_label": "188" }
6.2 The Prime Number Theorem 189 (d) For 0 ≤ κ ≤ 1 let Nκ(x ) denote the number of n ≤ x such that a(n) ≤ nκ. Show that Nκ(x ) = ∑ a≤x κ a∈A ∑ a1/κ−1 ≤b≤x /a b∈B 1. (e) Show that if κ is ﬁxed, 0 ≤ κ ≤ 1, then Nk (x ) = c(κ)x + O ( x√ log x ) where c(κ) = 1 π ∫ κ 0 du√u(1 − u) . 23. The deﬁnition of li( x ) is somewhat arbitrary because of the casual choice of the lower endpoint of integration. A more intrinsic logarithmic integral is Li(x ), which is deﬁned to be Li(x ) = lim ε→0+ (∫ 1−ε 0 + ∫ x 1+ε )dt log t (6.20) for x > 1. (Note that li( x ) = Li(x ) − Li(2).) (a) Show that ∫ 1−ε 0 dt log t =− ∫ ∞ − log(1−ε) e−v d v v . (b) Show that ∫ 1−ε 0 dt log t = log ε− ∫ ∞ 0 (log v)e−v d v + O (εlog 1 /ε), and explain why the integral on the right is Ŵ′(1) =− C0 . (c) Show that if x > 1, then ∫ x 1+ε dt log t = ∫ log x log(1+ε) ev d v v . (d) Show that if x > 1, then ∫ x 1+ε dt log t = log log x − log ε+ ∫ log x 1 ev − 1 v d v + O (ε). (e) Show that if x > 1, then Li(x ) = log log x + C0 + ∫ log x 0 ev − 1 v d v.	{ "creation_date": "2025-02-03", "file_name": "(Cambridge Studies in Advanced Mathematics 97) Hugh L. Montgomery, Robert C. Vaughan - Multiplicative number theory I_ Classical theory-Cambridge University Press (2006).pdf", "file_path": "/home/suryaremanan/Documents/IMO/augmentoolkit/original/input/(Cambridge Studies in Advanced Mathematics 97) Hugh L. Montgomery, Robert C. Vaughan - Multiplicative number theory I_ Classical theory-Cambridge University Press (2006).pdf", "file_size": 2365282, "file_type": "application/pdf", "last_modified_date": "2025-02-03", "page_label": "189" }
190 The Prime Number Theorem (f) Expand ev as a power series, and integrate term-by-term, to show that if x > 1, then Li(x ) = log log x + C0 + ∞∑ n=1 (log x )n n!n . (6.21) 24. For 0 < x < 1 let Li(x ) = ∫ x 0 dt log t . (a) Show that if 0 < x < 1, then Li(x ) = x log log 1 /x − ∫ ∞ − log x e−v log vd v. (b) Show that if 0 < x < 1, then Li(x ) = x log log 1 /x + C0 + ∫ − log x 0 e−v log vd v. (c) Show that if 0 < x < 1, then Li(x ) = log log 1 /x + C0 − ∫ − log x 0 1 − e−v v d v. (d) Show that if 0 < x < 1, then Li(x ) = log log 1 /x + C0 + ∞∑ n=1 (log x )n n!n . (e) (P ´ olya & Szeg ¨ o 1972, p. 8) Show that ∞∑ n=1 zn n!n =− ez ∞∑ n=1 (n∑ k=1 1 k ) (−z)n n! . (f) Show that if 0 < x < 1, then Li(x ) = log log 1 /x + C0 − x ∞∑ n=1 (n∑ k=1 1 k ) (log 1 /x )n n! . (6.22) 25. By repeated integration by parts we know that Li(x ) = x K∑ k=1 (k − 1)! (log x )k + OK ( x (log x )K +1 ) . Our object is to determine how closely one can approximate to Li( x )b y	{ "creation_date": "2025-02-03", "file_name": "(Cambridge Studies in Advanced Mathematics 97) Hugh L. Montgomery, Robert C. Vaughan - Multiplicative number theory I_ Classical theory-Cambridge University Press (2006).pdf", "file_path": "/home/suryaremanan/Documents/IMO/augmentoolkit/original/input/(Cambridge Studies in Advanced Mathematics 97) Hugh L. Montgomery, Robert C. Vaughan - Multiplicative number theory I_ Classical theory-Cambridge University Press (2006).pdf", "file_size": 2365282, "file_type": "application/pdf", "last_modified_date": "2025-02-03", "page_label": "190" }
6.2 The Prime Number Theorem 191 partial sums of the formal asymptotic expansion Li(x ) ∼ x ∞∑ k=1 (k − 1)! (log x )k . (a) Show that the least term in the sum above occurs when k = [log x ] + 1. (b) Show that if x ≥ e K , then Li(x ) = x K∑ k=1 (k − 1)! (log x )k + Li(e) + K −1∑ k=1 ( k! ∫ ek+1 ek dt (log t )k+1 − (k − 1)!ek kk ) − (K − 1)!e K K K + K ! ∫ x e K dt (log t )K +1 . (c) Deﬁne R(x ) by the relation Li(x ) = x [log x ]∑ k=1 (k − 1)! (log x )k + R(x ). Show that R(x ) is increasing, continuous, and convex downward for x ∈ [e K ,e K +1 ). Let αK = R(e K ), and let βK be the limit of R(x )a s x tends to e K +1 from below . (d) Show that ∫ e K +1 e K dt (log t )K +1 = e K K K ∫ 1/K 0 e K w (1 + w)K +1 d w. (e) Show that the integrand on the right above is ≤ 1 in the range of inte- gration. (f) Show that the minimum of e K w/(1 + w)K +1 for w> 0 occurs when w = 1/K . (g) Show that e K +1 (K + 1)K +1 < ∫ e K +1 e K dt (log t )K +1 < e K K K +1 . (h) Show that αK ր and that βK ց . (i) Show that βK − αK ≪ K −1/2 (j) Show that R(x ) = c + O ((log x )−1/2 ) where c = Li(e) + ∞∑ k=1 ( k! ∫ ek+1 ek dt (log t )k+1 − (k − 1)!ek kk ) .	{ "creation_date": "2025-02-03", "file_name": "(Cambridge Studies in Advanced Mathematics 97) Hugh L. Montgomery, Robert C. Vaughan - Multiplicative number theory I_ Classical theory-Cambridge University Press (2006).pdf", "file_path": "/home/suryaremanan/Documents/IMO/augmentoolkit/original/input/(Cambridge Studies in Advanced Mathematics 97) Hugh L. Montgomery, Robert C. Vaughan - Multiplicative number theory I_ Classical theory-Cambridge University Press (2006).pdf", "file_size": 2365282, "file_type": "application/pdf", "last_modified_date": "2025-02-03", "page_label": "191" }
192 The Prime Number Theorem (k) Show that if x ≥ e, then α1 ≤ Li(x ) − x [log x ]∑ k=1 (k − 1)! (log x )k ≤ β1 (6.23) where α1 =− 0.82316 ... and β1 = 1.259706 ... .. 26. (Ingham 1932, pp. 60–63) Suppose that η(t ) is deﬁned for t ≥ 2, that η′(t )i s continuous, η′(t ) → 0a s t →∞ , that η(t ) ց, that 1 /log t ≪ η(t ) ≤ 1/2, and that ζ(s) ̸=0 for σ ≥ 1 − η(t ), t ≥ 2. For x ≥ 2, put ω(x ) = min 2≤t <∞ η(t ) log x + log t . (a) Show that there is an absolute constant c > 0 such that π(x ) = li(x ) + O (x exp(−cω(x ))). (b) Show that if a > 0 is ﬁxed and (6.24) below holds, then (6.27) below holds with b = 1/(1 + a). (c) Show that (6.28) follows from (6.26). 6.3 Notes Section 6.1. Jensen (1899) proved that if f satisﬁes the hypotheses of Lemma 6.1, then \| f (0)\| n∏ k=1 R \|zk \| = exp (1 2π ∫ 2π 0 log \| f ( Re i θ)\| d θ ) where z1 ,..., zn are the zeros of f in the disc \|z\|≤ R. Here the right-hand side may be regarded as being the geometric mean of \| f (z)\| for z on the circle \|z\|= R. Each factor of the product above is ≥ 1, and if \|zk \|≤ r , then R/\|zk \|≥ R/r . Thus Lemma 6.1 follows easily from the above. The products used in the proofs of Lemmas 6.1 and 6.3 are known as Blaschke products. Their use (usually with inﬁnitely many factors) is an important tool of complex analysis. Lemma 6.2 is due to Borel (1897); it reﬁnes an earlier estimate of Hadamard. Carath´ eodory’s contributions on this subject are recounted by Landau (1906; Section 4). Lemma 6.4 is implicit in Landau (1909, p. 372), and may have been known earlier. It can also be easily derived from the identity (10.29) that arises by applying Hadamard’s theory of entire functions to the zeta function. The Prime Number Theorem was ﬁrst proved, in the qualitative form π(x ) ∼ x /log x , independently by Hadamard (1896) and de la V all´ ee Poussin (1896). In these papers, it was shown that ζ(1 + it ) ̸=0, but no speciﬁc zero-free region	{ "creation_date": "2025-02-03", "file_name": "(Cambridge Studies in Advanced Mathematics 97) Hugh L. Montgomery, Robert C. Vaughan - Multiplicative number theory I_ Classical theory-Cambridge University Press (2006).pdf", "file_path": "/home/suryaremanan/Documents/IMO/augmentoolkit/original/input/(Cambridge Studies in Advanced Mathematics 97) Hugh L. Montgomery, Robert C. Vaughan - Multiplicative number theory I_ Classical theory-Cambridge University Press (2006).pdf", "file_size": 2365282, "file_type": "application/pdf", "last_modified_date": "2025-02-03", "page_label": "192" }
6.3 Notes 193 was established. The ﬁrst proof that ζ(1 + it ) ̸=0 given by de la V all´ ee Poussin was rather complicated, but later in his long paper he gave a second proof depending on the inequality 1− cos 2 θ ≤ 4(1 + cos θ). This is equivalent to the non-negativity of the cosine polynomial 3 + 4 cos θ + cos 2 θ, which Mertens (1898) used to obtain the result of Exercise 6.4. Our Lemma 6.5 is derived by the same method. The classical zero-free region of Theorem 6.6 was established ﬁrst by de la V all´ ee Poussin (1899). The estimates (6.6) and (6.8) of Theorem 6.7 were ﬁrst proved by Gronwall (1913). Wider zero-free regions have been established by using exponential sum es- timates to obtain better upper bounds for \|ζ(s)\| when σ is near 1 . The ﬁrst such improvement was derived by Hardy & Littlewood. Their paper on this was never published, but accounts of their approach have been given by Landau (1924b) and Titchmarsh (1986, Chapter 5). Littlewood (1922) announced that from these estimates he had deduced thatζ(s) ̸=0 for σ ≥ 1 − c(log log τ)/log τ. As explained by Ingham (1932, p. 66), Littlewood never published his com- plicated proof, because the simpler method of Landau (1924a) had become available. In 1935, V inogradov introduced a new method for estimating W eyl sums. A W eyl sum is a sum of the form ∑ N n=1 e( f (n)) where f ∈ R[x ]. The quality of V inogradov’s estimate depends on rational approximations to the coefﬁcients off , and on the degree of f . The function f (x ) = t log x is not a polynomial, but by approximating to it by polynomials one can make V inogradov’s method apply . This was ﬁrst done by Chudakov (1936 a, b, c), who derived estimates forζ(s) for σ near 1 that allowed him to deduce that ζ(s) ̸=0 for σ> 1 − c(log τ)−a (6.24) for a > 10/11. V inogradov (1936b) gave stronger exponential sum estimates, which Titchmarsh (1938) used to obtain a zero-free region of the above form for a > 4/5. Hua (1949) introduced a further reﬁnement of V inogradov’s method, from which Titchmarsh (1951, Chapter 6) and T atuzawa (1952) derived the zero-free region σ> 1 − c(log τ)−3/4 (log log τ)−3/4 . By reﬁning the passage from W eyl sums to the zeta function, Korobov (1958a) obtained (6.24) fora > 5/7, and then Korobov (1958b, c) and V inogradov (1958) obtained a > 2/3. In fact, V inogradov claimed that one can take a = 2/3, but this seems to be still out of reach. Richert’s polished exposition of V inogradov’s method is reproduced in W alﬁsz (1963). Other expositions have since been given by Karatsuba & V oronin (1992, Chapter 4), Montgomery (1994, Chapter 4), and V aughan (1997). Richert (1967) used V inogradov’s	{ "creation_date": "2025-02-03", "file_name": "(Cambridge Studies in Advanced Mathematics 97) Hugh L. Montgomery, Robert C. Vaughan - Multiplicative number theory I_ Classical theory-Cambridge University Press (2006).pdf", "file_path": "/home/suryaremanan/Documents/IMO/augmentoolkit/original/input/(Cambridge Studies in Advanced Mathematics 97) Hugh L. Montgomery, Robert C. Vaughan - Multiplicative number theory I_ Classical theory-Cambridge University Press (2006).pdf", "file_size": 2365282, "file_type": "application/pdf", "last_modified_date": "2025-02-03", "page_label": "193" }
194 The Prime Number Theorem method to show that ζ(s) ≪ t 100(1−σ)3/2 (log t )2/3 (6.25) for σ ≤ 1, t ≥ 2. From this it follows that ζ(s) ̸=0 for σ ≥ 1 − c(log τ)−2/3 (log log τ)−1/3 . (6.26) The methods of Hadamard and de la V all´ ee Poussin depended on the analytic continuation ofζ(s), on bounds for the size of ζ(s) in the complex plane, and on Hadamard’s theory of entire functions. The ﬁrst two of these are achieved most easily by Riemann’s functional equation (see Corollaries 10.3–10.5). An abbreviated account of the third is found in Lemma 10.11. Landau (1903a) showed that one can obtain a zero-free region using only the local analytic properties of the zeta function. This enabled Landau to prove the Prime Ideal Theorem, which is the natural extension of the Prime Number Theorem to algebraic number ﬁelds: IfK is an algebraic number ﬁeld, then the number of prime ideals p in K with N (p) ≤ x is asymptotic to x /log x as x →∞ . This could not have been done at that time by the methods of Hadamard and de la V all´ ee Poussin, since the analytic continuation and functional equation of the Dedekind zeta functionζK (s) was established only later, by Hecke (1917). Landau did not achieve Theorem 6.6 at the ﬁrst attempt, but he reﬁned his approach in a series of papers culminating in the polished exposition of Landau (1924a). Section 6.2. Ingham (1932, pp. 60–65; cf. Titchmarsh 1986, pp. 56–60) developed a general system by which any given zero-free region of the zeta function can be used to derive an associated bound for the error term in the Prime Number Theorem. In particular, he showed that ifζ(s) ̸=0 for s in the region (6.24), then ψ(x ) = x + O (x exp(−c(log x )b )) (6.27) where b = 1/(1 + a). Similarly , from the zero-free region (6.26) it follows that π(x ) = li(x ) + O ( x exp ( − c(log x )3/5 (log log x )−1/5 )) . (6.28) Tur ´an (1950) used his method of power sums to show conversely that (6.27) implies (6.24). More general converse theorems have since been established by St´ as (1961) and Pintz (1980, 1983, 1984). A similar converse theorem in which an upper bound forM (x ) = ∑ n≤x µ(n) is used to produce a zero-free region has been given by Allison (1970). That M (x ) = o(x ) was ﬁrst proved by von Mangoldt (1897). The quantitative estimate (6.17) is due to Landau (1908a). The relation (6.19), asserted by Euler	{ "creation_date": "2025-02-03", "file_name": "(Cambridge Studies in Advanced Mathematics 97) Hugh L. Montgomery, Robert C. Vaughan - Multiplicative number theory I_ Classical theory-Cambridge University Press (2006).pdf", "file_path": "/home/suryaremanan/Documents/IMO/augmentoolkit/original/input/(Cambridge Studies in Advanced Mathematics 97) Hugh L. Montgomery, Robert C. Vaughan - Multiplicative number theory I_ Classical theory-Cambridge University Press (2006).pdf", "file_size": 2365282, "file_type": "application/pdf", "last_modified_date": "2025-02-03", "page_label": "194" }
6.4 References 195 (1748; Chapter 15, no. 277), was ﬁrst proved by von Mangoldt (1897). Landau (1899a) and de la V all´ee Poussin (1899) shortly gave simpler proofs. 6.4 References Allison, D. (1970). On obtaining zero-free regions for the zeta-function from estimates of M (x ), Proc. Cambridge Philos. Soc. 67, 333–337. Borel, E. (1897). Sur les z´ eros des fonctions enti` ers, Acta Math. 20, 357–396. Chudakov , N. G. (1936a). Sur les z´ eros de la fonction ζ(s), C. R. Acad. Sci. Paris 202, 191–193. (1936b). On zeros of the function ζ(s), Dokl. Akad. Nauk SSSR 1, 201–204. (1936c). On zeros of Dirichlet’s L -functions, Mat. Sb. (1) 43, 591–602. (1937). On W eyl’s sums, Mat. Sb. (2) 44, 17–35. (1938). On the functions ζ(s) and π(x ), Dokl. Akad. Nauk SSSR 21, 421–422. Cipolla, M. (1902). La determinazione assintotica dell’ nimo numero primo, Rend. Accad. Sci. Fis-Mat. Napoli (3) 8, 132–166. Euler, L. (1748). Introductio in analysin inﬁnitorum , I, Lausanne; Opera omnia Ser 1, V ol. 8, T eubner, 1922. Gronwall, T . H. (1913). Sur la fonction ζ(s) de Riemann au voisinage de σ = 1, Rend. Mat. Cir . P alermo 35, 95–102. Hadamard, J. (1896). Sur la distribution des z´ eros de la fonction ζ(s) et ses cons´ equences arithm´ etiques,Bull. Soc. Math. France 24, 199–220. Hardy , G. H. (1921). Note on Ramanujan’s trigonometrical function cq (n), and certain series of arithmetical functions, Proc. Cambridge Philos. Soc. 20, 263–271. Hecke, E. (1917). ¨Uber die Zetafunktion beliebiger algebraischer Zahlk ¨ orper, Nachr . Akad. Wiss. G ¨ottingen, 77–89; Mathematische W erke,G ¨ottingen: V andenhoeck & Ruprecht, 1959, pp. 159–171. Hua, L. K. (1949). An improvement of V inogradov’s mean-value theorem and several applications, Quart. J. Math. Oxford Ser . 20, 48–61. Ingham, A. E. (1932). The Distribution of Prime Numbers, Cambridge Tracts Math . 30. Cambridge: Cambridge University Press. (1945). Some T auberian theorems connected with the Prime Number Theorem, J. London Math. Soc. 20, 171–180. Jensen, J. L. W . V . (1899). Sur un nouvel et important th´ eor` eme de la th´ eorie des fonctions, Acta Math. 22, 359–364. Karatsuba, A. A. & V oronin, S. M. (1992). The Riemann Zeta-function . Berlin: de Gruyter. Korobov , N. M. (1958a). On the zeros of the function ζ(s), Dokl. Akad. Nauk SSSR 118, 231–232. (1958b). W eyl’s estimates of sums and the distribution of primes, Dokl. Akad. Nauk SSSR 123, 28–31. (1958c). Evaluation of trigonometric sums and their applications, Usp. Mat. Nauk 13, no. 4, 185–192. Landau, E. (1899a). Neuer Beweis der Gleichung ∑ ∞ k=1 µ(k) k = 0, Inaugural Dissertation, Berlin; Collected W orks, V ol. 1. Essen: Thales V erlag, pp. 69–83.	{ "creation_date": "2025-02-03", "file_name": "(Cambridge Studies in Advanced Mathematics 97) Hugh L. Montgomery, Robert C. Vaughan - Multiplicative number theory I_ Classical theory-Cambridge University Press (2006).pdf", "file_path": "/home/suryaremanan/Documents/IMO/augmentoolkit/original/input/(Cambridge Studies in Advanced Mathematics 97) Hugh L. Montgomery, Robert C. Vaughan - Multiplicative number theory I_ Classical theory-Cambridge University Press (2006).pdf", "file_size": 2365282, "file_type": "application/pdf", "last_modified_date": "2025-02-03", "page_label": "195" }
196 The Prime Number Theorem (1899b). Contribution ` al at h ´ eorie de la fonction ζ(s) de Riemann, C. R. Acad. Sci. Paris, 129, 812–815; Collected W orks , V ol. 1. Essen: Thales V erlag, 1985, pp. 84–88. (1900). Sur quelques probl` emes r´ elatifs ` a la distribution des nombres premiers, Bull. Soc. Math. France 28, 25–38; Collected W orks, V ol. 1. Essen: Thales V erlag, 1985, pp. 92–105. (1901a). ¨Uber die asymptotischen W erthe einiger zahlentheoretischer Functionen, Math. Ann. 54, 570–591; Collected W orks, V ol. 1. Essen: Thales V erlag, 1985, pp. 141–162. (1901b). Solutions de questions propos´ ees, Nouv . Ann. de Math. (4) 1, 281–283; Collected W orks, V ol. 1. Essen: Thales V erlag, 1985, pp. 181–182. (1903a). Neuer Beweis des Primzahlsatzes und Beweis des Primidealsatzes, Math. Ann. 56, 645–670; Collected W orks, V ol. 1. Essen: Thales V erlag, 1985, pp. 327– 353. (1903b). ¨Uber die Maximalordnung der Permutationen gegebenen Grades, Arch. Math. Phys. (3) 5, 92–103; Collected W orks , V ol. 1. Essen: Thales V erlag, 1985, pp. 384–396. (1903c). ¨Uber die zahlentheoretische Funktion µ(k), Sitzungsber . Kaiserl. Akad. Wiss. Wien math-natur . Kl. 112, 537–570; Collected W orks, V ol. 2. Essen: Thales V erlag, 1986, pp. 60–93. (1905). Sur quelques in´ egalit´es dans la th´ eorie de la fonction ζ(s) de Riemann, Bull. Soc. Math. France 33, 229–241; Collected W orks , V ol. 2. Essen: Thales V erlag, 1986, pp. 167–179. (1906). ¨Uber den Picardschen Satz, V ierteljahrschr . der Naturf. Ges. Z ¨urich 51, 252– 318; Collected W orks, V ol. 3. Essen: Thales V erlag, 1986, pp. 113–179. (1907). ¨Uber die Multiplikation Dirichlet’scher Reihen, Rend. Circ. Mat. P alermo 24, 81–160; Collected W orks, V ol. 3. Essen: Thales V erlag, 1986, pp. 323–401. (1908a). Beitr¨ age zur analytischen Zahlentheorie, Rend. Mat. Circ. P alermo 26, 169– 302; Collected W orks, V ol. 3. Essen: Thales V erlag, 1986, pp. 411–544. (1908b). ¨Uber die Einteilung der positiven ganzen Zahlen in vier Klassen nach der Mindestzahl der zu ihrer additiven Zusammensetzung erforderlichen Quadrate, Arch. Math Phys.(3) 13, 305–312; Collected W orks, V ol. 4. Essen: Thales V erlag, 1986, 59–66. (1909). Handbuch der Lehre von der V erteilung der Primzahlen , Leipzig: T eubner. (1924a). ¨Uber die Wurzeln der Zetafunktion, Math. Z. 20, 98–104; Collected W orks, V ol. 8. Essen: Thales V erlag, 1987, pp. 70–76. (1024b). ¨Uber die ζ-funktion und die L -funktionen, Math. Z. 20, 105–125; Collected W orks, V ol. 8. Essen: Thales V erlag, 1987, pp. 77–98. Littlewood, J. E. (1922). Researches in the theory of the Riemann ζ-function, Proc. London Math. Soc. (2), 20, xxii–xxvii; Collected papers , V ol. 2. Oxford: Oxford University Press, 1982, pp. 844–850. von Mangoldt, H. (1897). Beweis der Gleichung ∑ ∞ k=1 µ(k) k = 0, Sitzungsber . K ¨onigl. Preuß. Akad. Wiss . Berlin, 835–852. Massias, J.-P ., Nicolas, J.-L., & Robin, G. (1988). ´Evaluation asymptotique de l’ordre maximum d’un ´ el´ ement du groupe sym´ etrique, Acta Arith. 50, 221–242. (1989). Effective bounds for the maximal order of an element in the symmetric group, Math. Comp. 53, 665–678.	{ "creation_date": "2025-02-03", "file_name": "(Cambridge Studies in Advanced Mathematics 97) Hugh L. Montgomery, Robert C. Vaughan - Multiplicative number theory I_ Classical theory-Cambridge University Press (2006).pdf", "file_path": "/home/suryaremanan/Documents/IMO/augmentoolkit/original/input/(Cambridge Studies in Advanced Mathematics 97) Hugh L. Montgomery, Robert C. Vaughan - Multiplicative number theory I_ Classical theory-Cambridge University Press (2006).pdf", "file_size": 2365282, "file_type": "application/pdf", "last_modified_date": "2025-02-03", "page_label": "196" }
6.4 References 197 Mertens, F . (1897). Ueber eine Zahlentheoretische Function, Sitzungsber . Akad. Wiss. Wien Abt. 2a 106. (1898). ¨Uber eine Eigenschaft der Riemannscher ζ-Funktion, Sitzungsber . Kais. Akad. Wiss. Wien Abt. 2a 107, 1429–1434. Montgomery , H. L. (1994). T en Lectures on the Interface Between Analytic Number The- ory and Harmonic Analysis , CBMS Regional Conf. Series in Math. 84. Providence: Amer. Math. Soc. Montgomery , H. L. & V aughan, R. C. (2001). Mean values of multiplicative functions, P eriod. Math. Hungar . 43, 199–214. Pintz, J. (1980). On the remainder term of the prime number formula, II. On a theorem of Ingham, Acta Arith. 37, 209–220. (1983). Oscillatory Properties of the Remainder T erm of the Prime Number Formula, Studies in Pure Math. Basel: Birkh¨ auser, pp. 551–560. (1984). On the remainder term of the prime number formula and the zeros of Rie- mann’s zeta-function, Number Theory (Noordwijkerhout, 1983). Lecture notes in math. 1068. Berlin: Springer-V erlag, pp. 186–197. P´olya, G. & Szeg ¨ o, G. (1972). Problems and Theorems in Analysis, V ol. 1. Grundl. math. Wiss. 193. New Y ork: Springer-V erlag. Richert, H.-E. (1967). Zur Absch¨ atzung der Riemannschen Zetakunktion in der N¨ ahe der V ertikalen σ = 1, Math. Ann. 169, 97–101. Rosser, J. B. (1939). The n-th prime is greater than n log n, Proc. London Math. Soc. (2) 45, 21–44. Rosser, J. B. & Schoenfeld, L. (1962). Approximate formulas for some functions of prime numbers, Illinois J. Math. 6, 64–94. St´ as, W . (1961). ¨Uber die Umkehrung eines Satzes von Ingham, Acta Arith. 6, 435– 446. T atuzawa, T . (1952). On the number of primes in an arithmetic progression, Jap. J. Math. 21, 93–111. Titchmarsh, E. C. (1938). On ζ(s) and π(x ), Quart. J. Math. Oxford Ser . 9, 97–108. (1951). The Theory of the Riemann Zeta-function , Oxford: Oxford University Press. (1986). The Theory of the Riemann Zeta-function , Second Ed. Oxford: Oxford University Press. Tur ´an, P . (1950). On the remainder-term in the prime-number formula, II, Acta. Math. Acad. Sci. Hungar . 1, 155–166; Collected P apers , V ol. 1. Budapest: Akad´ emiai Kiado, 1990, pp. 541–551. de la V all´ ee Poussin, C. J. (1896). Recherches analytiques sur la th´ eorie des nombres premiers, I–III, Ann. Soc. Sci. Bruxelles 20, 183–256, 281–362, 363–397. (1899). Sur la fonction ζ(s) et le nombre des nombres premiers inf´ erieurs ` a une limite donn´ ee,Mem. Couronn ´es de l’Acad. Roy. Sci. Bruxelles 59. V aughan, R. C. (1997). The Hardy–Littlewood Method , Second Edition, Cambridge Tracts in Math. 125, Cambridge: Cambridge University Press. V inogradov , I. M. (1935). On W eyl’s sums, Mat. Sb. 42, 521–530. (1936a). A new method for resolving certain general questions in the theory of num- bers, Mat. Sb. (1) 43, 9–19. (1936b). A new method of estimation of trigonometrical sums, Mat. Sb. (1) 43, 175– 188.	{ "creation_date": "2025-02-03", "file_name": "(Cambridge Studies in Advanced Mathematics 97) Hugh L. Montgomery, Robert C. Vaughan - Multiplicative number theory I_ Classical theory-Cambridge University Press (2006).pdf", "file_path": "/home/suryaremanan/Documents/IMO/augmentoolkit/original/input/(Cambridge Studies in Advanced Mathematics 97) Hugh L. Montgomery, Robert C. Vaughan - Multiplicative number theory I_ Classical theory-Cambridge University Press (2006).pdf", "file_size": 2365282, "file_type": "application/pdf", "last_modified_date": "2025-02-03", "page_label": "197" }
198 The Prime Number Theorem (1947). The Method of Trigonometrical Sums in the Theory of Numbers , Trav . Inst. Math. Stecklov 23; English translation, London: Interscience Publishers, 1954. (1958). A new evaluation of ζ(1 + it ), Izv . Akad. Nauk SSSR 22, 161–164. W alﬁsz, A. (1963). W eylsche Exponentialsummen in der neuren Zahlentheorie , Math. Forschungsberichte 15. Berlin: Deutscher V erlag Wiss.	{ "creation_date": "2025-02-03", "file_name": "(Cambridge Studies in Advanced Mathematics 97) Hugh L. Montgomery, Robert C. Vaughan - Multiplicative number theory I_ Classical theory-Cambridge University Press (2006).pdf", "file_path": "/home/suryaremanan/Documents/IMO/augmentoolkit/original/input/(Cambridge Studies in Advanced Mathematics 97) Hugh L. Montgomery, Robert C. Vaughan - Multiplicative number theory I_ Classical theory-Cambridge University Press (2006).pdf", "file_size": 2365282, "file_type": "application/pdf", "last_modified_date": "2025-02-03", "page_label": "198" }
7 Applications of the Prime Number Theorem W e now use the Prime Number Theorem, and other estimates obtained by similar methods, to estimate the number of integers whose multiplicative structure is of a speciﬁed type. 7.1 Numbers composed of small primes Let ψ(x ,y) denote the number of integers n,1 ≤ n ≤ x , all of whose prime factors are ≤ y. Obviously , if y ≥ x , then ψ(x ,y) = [x ] = x + O (1). (7.1) Also, if n ≤ x , then n can have at most one prime factor p > √ x , and hence if x 1/2 ≤ y ≤ x , then ψ(x ,y) = [x ] − ∑ y<p≤x ∑ n≤x p\|n 1 = [x ] − ∑ y<p≤x [x /p] = x − x ∑ y<p≤x 1 p + O (π(x )). By the estimates of Chebyshev and Mertens (Corollary 2.6 and Theorem 2.7(d)), this is = x ( 1 − log log x log y ) + O (x log x ) . Thus if we take u = (log x )/(log y), so that y = x 1/u , then we see that ψ ( x ,x 1/u ) = (1 − log u)x + O (x log x ) (7.2) 199	{ "creation_date": "2025-02-03", "file_name": "(Cambridge Studies in Advanced Mathematics 97) Hugh L. Montgomery, Robert C. Vaughan - Multiplicative number theory I_ Classical theory-Cambridge University Press (2006).pdf", "file_path": "/home/suryaremanan/Documents/IMO/augmentoolkit/original/input/(Cambridge Studies in Advanced Mathematics 97) Hugh L. Montgomery, Robert C. Vaughan - Multiplicative number theory I_ Classical theory-Cambridge University Press (2006).pdf", "file_size": 2365282, "file_type": "application/pdf", "last_modified_date": "2025-02-03", "page_label": "199" }
200 Applications of the Prime Number Theorem 0 1 1 Figure 7.1 The Dickman function ρ(u) for 0 ≤ u ≤ 4. uniformly for 1 ≤ u ≤ 2. W e shall show more generally that there is a function ρ(u) > 0 such that ψ ( x ,x 1/u ) ∼ ρ(u)x (7.3) as x →∞ with u bounded. The function ρ(u) that arises here is known as the Dickman function ; it may be deﬁned to be the unique continuous function on [0,∞) satisfying the differential–delay equation uρ′(u) =− ρ(u − 1) (7.4) for u > 1 together with the initial condition that ρ(u) = 1 (7.5) for 0 ≤ u ≤ 1. Before proceeding further we note some simple properties of this function. By dividing both sides of (7.4) by u and then integrating, we ﬁnd that ρ(v) = ρ(u) − ∫ v u ρ(t − 1) dt t (7.6) for 1 ≤ u ≤ v. Also, from (7.4) we see that ( uρ(u))′ = ρ(u) − ρ(u − 1), so that by integrating it follows that uρ(u) = ∫ u u−1 ρ(v) d v + C for u ≥ 1, where C is a constant of integration. On taking u = 1 we deduce that C = 0, and hence that uρ(u) = ∫ u u−1 ρ(v) d v (7.7) for u ≥ 1. As might be surmised from Figure 7.1, ρ(u) is positive and decreasing. T o prove this, let u0 be the inﬁmum of the set of all solutions of the equation ρ(u) = 0. By the continuity of ρ it follows that ρ(u0 ) = 0. But ρ(u) > 0 for	{ "creation_date": "2025-02-03", "file_name": "(Cambridge Studies in Advanced Mathematics 97) Hugh L. Montgomery, Robert C. Vaughan - Multiplicative number theory I_ Classical theory-Cambridge University Press (2006).pdf", "file_path": "/home/suryaremanan/Documents/IMO/augmentoolkit/original/input/(Cambridge Studies in Advanced Mathematics 97) Hugh L. Montgomery, Robert C. Vaughan - Multiplicative number theory I_ Classical theory-Cambridge University Press (2006).pdf", "file_size": 2365282, "file_type": "application/pdf", "last_modified_date": "2025-02-03", "page_label": "200" }
7.1 Numbers composed of small primes 201 0 ≤ u < u0 , and hence if we take u = u0 in (7.7), then the left-hand side is 0 while the right-hand side is positive, a contradiction. Thus ρ(u) > 0 for all u ≥ 0, and by (7.4) it follows that ρ′(u) < 0 for all u > 1. Figure 7.1 also suggests that ρ(u) tends to 0 rapidly as u →∞ . W e now establish a crude estimate in this direction. Lemma 7.1The function ρ(u) is positive and decreasing for u ≥ 0, and satisﬁes the inequalities 1 2Ŵ(2u + 1) ≤ ρ(u) ≤ 1 Ŵ(u + 1) . Proof For positive integers U we prove by induction that the upper bound holds for 0 ≤ u ≤ U . T o provide the basis of the induction we need to show that Ŵ(s) ≤ 1 for 1 ≤ s ≤ 2. This is immediate from the relations Ŵ(1) = Ŵ(2) = 1,Ŵ ′′(s) = ∫ ∞ 0 e−x x s−1 (log x )2 dx > 0( 0 < s < ∞). (7.8) Since ρ(u) is decreasing, we see by (7.7) that uρ(u) ≤ ρ(u − 1). Thus if the desired upper bound holds for u ≤ U and if U ≤ u ≤ U + 1, then ρ(u) ≤ ρ(u − 1) u ≤ 1 uŴ(u) = 1 Ŵ(u + 1) by (C.4). After making the change of variables u = v/2, the desired lower bound asserts that ρ(v/2) ≥ 1/(2Ŵ(v + 1)). W e let V run through positive integral values, and prove by induction on V that the lower bound holds for 0 ≤ v ≤ V . T o establish the lower bound for 0 ≤ v ≤ 2 it sufﬁces to show that Ŵ(s) ≥ 1/2 for all s > 0. From (7.8) we see that Ŵ(s) ≥ 1 for 0 < s ≤ 1 and for s ≥ 2; thus it remains to note that if 1 ≤ s ≤ 2, then Ŵ(s) = ∫ ∞ 0 e−x x s−1 dx ≥ ∫ 1 0 e−x xd x + ∫ ∞ 1 e−x dx = 1 − 1 e > 1 2 . (The actual fact of the matter is that min s>0 Ŵ(s) = Ŵ(1.4616 ... ) = 0.8856 ... .) Since ρ(u) is decreasing, we see by (7.7) that uρ(u) ≥ ρ(u − 1/2)/2. Thus if the lower bound holds for 0 ≤ v ≤ V and if V ≤ v ≤ V + 1, then ρ(v/2) ≥ ρ((v − 1)/2) v ≥ 1 2vŴ(v) = 1 2Ŵ(v + 1) by (C.4). This completes the inductive step, so the proof is complete. □ W e now use elementary reasoning to show that (7.3) holds uniformly for u in bounded intervals.	{ "creation_date": "2025-02-03", "file_name": "(Cambridge Studies in Advanced Mathematics 97) Hugh L. Montgomery, Robert C. Vaughan - Multiplicative number theory I_ Classical theory-Cambridge University Press (2006).pdf", "file_path": "/home/suryaremanan/Documents/IMO/augmentoolkit/original/input/(Cambridge Studies in Advanced Mathematics 97) Hugh L. Montgomery, Robert C. Vaughan - Multiplicative number theory I_ Classical theory-Cambridge University Press (2006).pdf", "file_size": 2365282, "file_type": "application/pdf", "last_modified_date": "2025-02-03", "page_label": "201" }
202 Applications of the Prime Number Theorem Theorem 7.2 (Dickman) Let ψ(x ,y) be the number of positive integers not exceeding x composed entirely of prime numbers not exceeding y, and let ρ(u) be deﬁned as above. Then for any U ≥ 0 we have ψ ( x ,x 1/u ) = ρ(u)x + O (x log x ) (7.9) uniformly for 0 ≤ u ≤ U and all x ≥ 2. Proof W e restrict U to integral values, and induct on U . The basis of the induction is provided by (7.1) and (7.5). Also, (7.2) gives (7.9) for 1 ≤ u ≤ 2 since from (7.6) we see that ρ(u) = 1 − log u (7.10) for 1 ≤ u ≤ 2. Suppose now that U is an integer, U ≥ 2, and that (7.9) holds uniformly for 0 ≤ u ≤ U . W e show that (7.9) holds uniformly for U ≤ u ≤ U + 1. T o this end we classify n according to the size of the largest prime factor P (n)o f n. Thus we see that ψ(x ,y) = 1 + ∑ p≤y card{n ≤ x : P (n) = p}. Here the ﬁrst term on the right reﬂects the fact that if x ≥ 1, then ψ(x ,y) counts the number n = 1 for which P (1) is undeﬁned. In the sum on the right, the summand is ψ(x /p, p), and hence we see that ψ(x ,y) = 1 + ∑ p≤y ψ(x /p, p). (7.11) On differencing, it follows that if y ≤ z, then ψ(x ,y) = ψ(x ,z) − ∑ y<p≤z ψ(x /p, p). (7.12) Suppose that z = x 1/U and that y = x 1/u with U ≤ u ≤ U + 1. Deﬁne u p by the relation p = (x /p)1/u p . That is, u p = log x log p − 1, which is ≤ u − 1 ≤ U if p ≥ y. Hence by the inductive hypothesis the right- hand side of (7.12) is ρ(U )x + O (x log x ) − x ∑ y<p≤z ρ((log x )/(log p) − 1) p + O ( x ∑ y<p≤z 1 p log x /p ) . (7.13)	{ "creation_date": "2025-02-03", "file_name": "(Cambridge Studies in Advanced Mathematics 97) Hugh L. Montgomery, Robert C. Vaughan - Multiplicative number theory I_ Classical theory-Cambridge University Press (2006).pdf", "file_path": "/home/suryaremanan/Documents/IMO/augmentoolkit/original/input/(Cambridge Studies in Advanced Mathematics 97) Hugh L. Montgomery, Robert C. Vaughan - Multiplicative number theory I_ Classical theory-Cambridge University Press (2006).pdf", "file_size": 2365282, "file_type": "application/pdf", "last_modified_date": "2025-02-03", "page_label": "202" }
7.1 Numbers composed of small primes 203 Let s(w) = ∑ p≤w 1/p, and write Mertens’ estimate (Theorem 2.7(d)) in the form s(w) = log log w+ c + r (w). Then the sum in the main term above is∫ z y ρ((log x )/(log w) − 1) ds (w) = ∫ z y ρ((log x )/(log w) − 1) d log log w + ∫ z y ρ((log x )/(log w) − 1) dr (w). (7.14) W e put t = (log x )/(log w). Since d log log w = d w wlog w =− dt t , the ﬁrst integral on the right-hand side of (7.14) is∫ u U ρ(t − 1) dt t . (7.15) By integrating by parts and the estimate r (w) ≪ 1/log wwe see that the second integral on the right-hand side of (7.14) is ρ((log x )/(log w) − 1)r (w) ⏐ ⏐ ⏐ ⏐ z y − ∫ z y r (w) dρ((log x )/(log w) − 1) ≪ 1 log x ( 1 + ∫ z y 1 \|dρ((log x )/(log w) − 1)\| ) ≪ 1 log x since ρ is monotonic and bounded. By Mertens’ estimate (Theorem 2.7(d)) we also see that the error term in (7.13) is ≪ x log x ∑ y<p≤z 1 p ≪ x log x since log log z = log log y + O (1). On combining our estimates in (7.12) we ﬁnd that ψ(x ,x 1/u ) = x ( ρ(U ) − ∫ u U ρ(t − 1) dt t ) + O (x log x ) . Thus by (7.6) we have the desired estimate for U ≤ u ≤ U + 1, and the proof is complete. □ As for ψ(x ,y) when y < x ε, we show next that ψ(x ,(log x )a ) = x 1−1/a+o(1) (7.16) for any ﬁxed a ≥ 1. The upper bound portion of this is obtained by means of bounds for an associated Dirichlet series, while the lower bound is derived by combinatorial reasoning.	{ "creation_date": "2025-02-03", "file_name": "(Cambridge Studies in Advanced Mathematics 97) Hugh L. Montgomery, Robert C. Vaughan - Multiplicative number theory I_ Classical theory-Cambridge University Press (2006).pdf", "file_path": "/home/suryaremanan/Documents/IMO/augmentoolkit/original/input/(Cambridge Studies in Advanced Mathematics 97) Hugh L. Montgomery, Robert C. Vaughan - Multiplicative number theory I_ Classical theory-Cambridge University Press (2006).pdf", "file_size": 2365282, "file_type": "application/pdf", "last_modified_date": "2025-02-03", "page_label": "203" }
204 Applications of the Prime Number Theorem An upper bound for ψ(x ,y) can be constructed by observing that if σ> 0, then ψ(x ,y) ≤ ∑ n≤x p\|n⇒ p≤y (x n )σ ≤ x σ ∑ p\|n⇒ p≤y 1 nσ = x σ ∏ p≤y ( 1 − 1 pσ )−1 .(7.17) Rankin used this chain of inequalities to derive an upper bound for ψ(x ,y). This approach is fruitful in a variety of settings, and has become known as ‘Rankin’s method’. T o use the above, we must establish an upper bound for the product on the right-hand side. The size of this product is a little difﬁcult to describe, because its behaviour depends on the size ofσ.I f σ is near 0, then most of the factors are ap- proximately (1 − y−σ)−1 , and hence we expect the product to be approximately (1 − y−σ)−y/log y .I f σ is larger (but still < 1), then the general factor is approx- imately exp( p−σ), and hence the product is approximately the exponential of ∑ p≤y p−σ ∼ ∫ y 2 dt t σ log t ∼ y1−σ (1 − σ) log y . W e begin by making these relations precise. Lemma 7.3If 0 ≤ σ ≤ 1, then ∑ p≤y p−σ = ∫ y 2 du uσ log u + O ( y1−σ exp ( − c √ log y )) + O (1). (7.18) Proof W e write the left-hand side as∫ y 2− u−σ d π(u) = ∫ y 2− u−σ d li(u) + ∫ y 2− u−σ dr (u) where r (u) = π(u) − li(u). The ﬁrst integral on the right is ∫y 2 u−σ(log u)−1 du . By integrating by parts we ﬁnd that the second integral is y−σr ( y) − 2−σr (2−) + σ ∫ y 2 r (u)u−σ−1 du . Suppose that b is a positive constant chosen so that r (u) ≪ u exp(−b√ log u). Then the ﬁrst two terms above can be absorbed into the error terms in (7.18) if c< b. T o complete the proof it sufﬁces to show that∫ y 2 u−σ exp(−b √ log u) du ≪ 1 + y1−σ exp ( − b 3 √ log y ) , (7.19) for then we have (7.18) with c = b/3. T o prove (7.19) we note that if σ ≥ 1 − b/(2√ log y), then u1−σ exp ( − b 2 √ log u ) = exp ( (1 − σ) log u − b 2 √ log u ) ≤ exp (b 2 (log u)/ √ log y − b 2 √ log u ) ≤ 1	{ "creation_date": "2025-02-03", "file_name": "(Cambridge Studies in Advanced Mathematics 97) Hugh L. Montgomery, Robert C. Vaughan - Multiplicative number theory I_ Classical theory-Cambridge University Press (2006).pdf", "file_path": "/home/suryaremanan/Documents/IMO/augmentoolkit/original/input/(Cambridge Studies in Advanced Mathematics 97) Hugh L. Montgomery, Robert C. Vaughan - Multiplicative number theory I_ Classical theory-Cambridge University Press (2006).pdf", "file_size": 2365282, "file_type": "application/pdf", "last_modified_date": "2025-02-03", "page_label": "204" }
7.1 Numbers composed of small primes 205 for 2 ≤ u ≤ y. Hence for σ in this range the integral in (7.19) is ≤ ∫ y 2 du u exp (b 2 √ log u )< ∫ ∞ 2 du u exp (b 2 √ log u )≪ 1. Now suppose that σ ≤ 1 − b 2√ log y . (7.20) W e write the integral in (7.19) as ∫y1/4 2 + ∫y y1/4 = I1 + I2 , say . Then I1 ≤ ∫ y1/4 2 u−σ du < y(1−σ)/4 1 − σ , which by (7.20) is ≪ y1−σ√ log y exp ( − 3 4 (1 − σ) log y ) ≪ y1−σ exp ( − b 3 √ log y ) . As for I2 , we note that if u ≥ y1/4 , then log u ≥ 1 4 log y. Hence I2 ≤ exp ( − b 2 √ log y )∫ y 2 u−σ du ≤ exp ( − b 2 √ log y )y1−σ 1 − σ ≪ exp ( − b 2 √ log y ) y1−σ√ log y ≪ y1−σ exp ( − b 3 √ log y ) . These estimates combine to give (7.19), so the proof is complete. □ Lemma 7.4 If y ≥ 2 and 1 − 4/log y ≤ σ ≤ 1, then ∑ p≤y p−σ = log log y + O (1). (7.21) If y ≥ 2 and 0 ≤ σ ≤ 1 − 4/log y, then ∑ p≤y p−σ = y1−σ (1 − σ) log y + log 1 1 − σ + O ( y1−σ (1 − σ)2 (log y)2 ) . (7.22) Proof Suppose that 1 − 4/log y ≤ σ ≤ 1. If u ≤ y, then u−σ = u−1 u1−σ = u−1 exp ( (1 − σ) log u ) = u−1 ( 1 + O ((1 − σ) log u) ) = u−1 + O ( u−1 (1 − σ) log u ) . Hence∫ y 2 du uσ log u = ∫ y 2 du u log u + O ( (1 − σ) ∫ y 2 du u ) = log log y + O (1). Thus (7.21) follows from Lemma 7.3. T o prove (7.22) we let v = exp(4/(1 − σ)), and observe that v ≤ y. W e write the integral in Lemma 7.3 as ∫v 2 + ∫y v = I1 + I2 , say . By the above we see that I1 = log log v + O (1) = log 1 /(1 − σ) + O (1). By integration by parts we see	{ "creation_date": "2025-02-03", "file_name": "(Cambridge Studies in Advanced Mathematics 97) Hugh L. Montgomery, Robert C. Vaughan - Multiplicative number theory I_ Classical theory-Cambridge University Press (2006).pdf", "file_path": "/home/suryaremanan/Documents/IMO/augmentoolkit/original/input/(Cambridge Studies in Advanced Mathematics 97) Hugh L. Montgomery, Robert C. Vaughan - Multiplicative number theory I_ Classical theory-Cambridge University Press (2006).pdf", "file_size": 2365282, "file_type": "application/pdf", "last_modified_date": "2025-02-03", "page_label": "205" }
206 Applications of the Prime Number Theorem that I2 = y1−σ (1 − σ) log y − v1−σ (1 − σ) log v + 1 1 − σ ∫ y v du uσ(log u)2 . Here the ﬁrst term on the right is one of the main terms in (7.22), and the second term isO (1). Let J denote the integral on the right. T o complete the proof it sufﬁces to show that J ≪ y1−σ (1 − σ)(log y)2 . (7.23) T o this end we integrate by parts again: J = y1−σ (1 − σ)(log y)2 − v1−σ (1 − σ)(log v)2 + 2 1 − σ ∫ y v d w wσ(log w)3 . Here the second term on the right-hand side is e4 2−4 (1 − σ) ≪ 1 − σ, while the ﬁrst term on the right-hand side is larger. As for the integral on the right, we observe that ifw ≥ v, then (log w)3 ≥ 4(log w)2 /(1 − σ). Hence the last term on the right above has absolute value not exceeding J /2. Thus we have (7.23), and the proof is complete. □ Lemma 7.5 Suppose that y ≥ 2.I f max ( 2/log y,1 − 4/log y ) ≤ σ ≤ 1, then ∏ p≤y ( 1 − p−σ)−1 ≍ log y. (7.24) If 2/log y ≤ σ ≤ 1 − 4/log y, then ∏ p≤y (1 − p−σ)−1 = 1 1 − σ × exp ( y1−σ (1 − σ) log y ( 1 + O ( 1 (1 − σ) log y ) + O ( y−σ) )) . (7.25) Proof The bound (7.24) is trivial when σ ≤ 2/3 since then y ≤ e12 . The estimate (1 − δ)−1 = exp ( δ + O (δ2 ) ) holds uniformly for \|δ\|≤ 1/2. W e take δ = p−σ for p >v = e1/σ to deduce that ∏ v<p≤y ( 1 − p−σ)−1 = exp (∑ v<p≤y p−σ + O (∑ v<p≤y p−2σ )) . Now (7.24) follows at once from Lemma 7.4 when σ ≥ 2/3. Thus it remains to establish (7.25). The sum in the error term above is ≪ 1 for σ> 5/8. If 3/8 ≤ σ ≤ 5/8, then by Lemma 7.4 it is ≪ y1/4 /log y.I f2 /log y ≤ σ ≤ 3/8, then by Lemma 7.4 the sum is ≪ y1−2σ/log y. Thus in any case this error term	{ "creation_date": "2025-02-03", "file_name": "(Cambridge Studies in Advanced Mathematics 97) Hugh L. Montgomery, Robert C. Vaughan - Multiplicative number theory I_ Classical theory-Cambridge University Press (2006).pdf", "file_path": "/home/suryaremanan/Documents/IMO/augmentoolkit/original/input/(Cambridge Studies in Advanced Mathematics 97) Hugh L. Montgomery, Robert C. Vaughan - Multiplicative number theory I_ Classical theory-Cambridge University Press (2006).pdf", "file_size": 2365282, "file_type": "application/pdf", "last_modified_date": "2025-02-03", "page_label": "206" }
7.1 Numbers composed of small primes 207 is majorized by the error terms on the right-hand side of (7.25). By Lemma 7.4, the main term is ∑ v<p≤y p−σ = y1−σ (1 − σ) log y + log 1 1 − σ + O ( y1−σ (1 − σ)2 (log y)2 ) + O (v log v ) . Since 2 /log y ≤ σ ≤ 1 − 4/log y, y satisﬁes y ≥ e6 , and σ(1 − σ) log y ≥ 2(1 − 2/log y) ≥ 4/3. Hence ( y1−σ)3/4 ≥ v and the second error term above is dominated by the ﬁrst. It remains to consider the contribution of the primes p ≤ v.I f σ> 1/3, then the contribution of these primes is ≪ 1, so we may suppose that 2 /log y ≤ σ ≤ 1/3. In this range 1 − p−σ ≍ σ log p = log p log v . Since ∑ p≤v log ( C log v log p ) ≪ v, it follows that ∏ p≤v (1 − p−σ)−1 < exp(C v) = exp ( Ce 1/σ) ≤ exp ( Cy 1/2 ) , which sufﬁces. Thus the proof is complete. □ W e now bound ψ(x ,y) by combining Lemma 7.5 with the inequalities (7.17). Theorem 7.6 If y = x 1/u and log x ≤ y ≤ x 1/9 , then ψ(x ,y) < x (log y)e x p ( − u log u − u log log u + u − u log log u log u + O (u log u ) + O (u2 log u y )) . Here the ﬁrst error term is larger than the second if y ≥ (log x ) log log x , while if y is smaller, then the second error term dominates. Proof W e ﬁrst note that we may suppose that y ≥ 9 log x , since the bound for smaller y follows by taking y = 9 log x . T o motivate the choice of σ in (7.17) we note that the expression to be minimized is approximately x σ exp (∫ y 2 u−σ log u du ) .	{ "creation_date": "2025-02-03", "file_name": "(Cambridge Studies in Advanced Mathematics 97) Hugh L. Montgomery, Robert C. Vaughan - Multiplicative number theory I_ Classical theory-Cambridge University Press (2006).pdf", "file_path": "/home/suryaremanan/Documents/IMO/augmentoolkit/original/input/(Cambridge Studies in Advanced Mathematics 97) Hugh L. Montgomery, Robert C. Vaughan - Multiplicative number theory I_ Classical theory-Cambridge University Press (2006).pdf", "file_size": 2365282, "file_type": "application/pdf", "last_modified_date": "2025-02-03", "page_label": "207" }
208 Applications of the Prime Number Theorem On taking logarithmic derivatives, this suggests that we should take σ to be the root of the equation log x = y1−σ 1 − σ. (7.26) In actual fact we take σ = 1 − log u + log log u log y . (7.27) It is easy to see that for this σ the right-hand side of (7.26) is log x log u log u + log log u , so it is reasonable to expect that the simple choice (7.27) is close enough to the root of (7.26) for our present purposes. From the inequalities 9 log x ≤ y ≤ x 1/9 it follows that the σ given by (7.27) satisﬁes 2 /log y ≤ σ ≤ 1 − 1/log y. Hence the stated upper bound follows by combining (7.17) with the estimates of Lemma 7.5. □ T o obtain companion lower bounds we observe that if k is chosen so that yk ≤ x , then ψ(x ,y) certainly counts all integers n composed of primes p ≤ y such that /Omega1 (n) ≤ k. Put r = π( y), and suppose that p1 , p2 ,..., pr are the primes not exceeding y. Then n is of the form n = pa1 1 pa2 2 ··· par r , and ψ(x ,y) is at least as large as the number of solutions of the inequality a1 + a2 +···+ ar ≤ k in non-negative integers ai . For this quantity we have an exact formula, as follows. Lemma 7.7 Let A (r,k) denote the number of solutions of the inequality a 1 + a2 +···+ ar ≤ k in non-negative integers a i . Then A (r,k) = (r +k k ) . Analytic Proof Let ar +1 = k − ∑ r i =1 ai . Then A(r,k) is the number of ways of writing k = a1 + a2 +···+ ar +1 , which is the coefﬁcient of x k in the power series (∞∑ a=0 x a )r +1 = (1 − x )−r −1 = ∞∑ k=0 (r + k k ) x k by the ‘negative’ binomial theorem. □ Combinatorial Proof Suppose that we have k circles ◦ and r bars \| arranged in a line. Let a1 be the number of circles to the left of the ﬁrst bar, let a2 be the number of circles between the ﬁrst and second bar, and so on, so that ar is the number of circles between the last two bars. (The number of circles to the right of the last bar isk − ∑ ai .) Thus a conﬁguration of circles and bars determines a choice of non-negative ai with a1 + a2 +···+ ar ≤ k. But conversely , a	{ "creation_date": "2025-02-03", "file_name": "(Cambridge Studies in Advanced Mathematics 97) Hugh L. Montgomery, Robert C. Vaughan - Multiplicative number theory I_ Classical theory-Cambridge University Press (2006).pdf", "file_path": "/home/suryaremanan/Documents/IMO/augmentoolkit/original/input/(Cambridge Studies in Advanced Mathematics 97) Hugh L. Montgomery, Robert C. Vaughan - Multiplicative number theory I_ Classical theory-Cambridge University Press (2006).pdf", "file_size": 2365282, "file_type": "application/pdf", "last_modified_date": "2025-02-03", "page_label": "208" }
7.1 Numbers composed of small primes 209 choice of such ai determines a conﬁguration of circles and bars. The number of ways of choosing the positions of the k circles in the r + k available places is (r +k k ) . □ Theorem 7.8 If log x ≤ y ≤ x , then ψ(x ,y) ≫ x y exp(−u log log x + u/2). Proof Let r = π( y) and let k be the largest integer such that yk ≤ x . That is, k = [u]. Then by Lemma 7.7 and Stirling’s formula we see that ψ(x ,y) ≥ (r + k k ) ≍ (r + k k )k (r + k r )r 1√ k . (7.28) The identity k log(1 + r/k) + r log(1 + k/r ) = ∫ r 0 log(1 + k/t ) dt shows that the left-hand side is an increasing function of r . It can be supposed that x is sufﬁciently large. Let z = y/(k log y). Then the expression (7.28) is ≫ ( 1 + y k log y )k ( 1 + k log y y )y/log y 1√ k ≥ (z(1 + 1/z)z )k , Moreover u − 1 < k ≤ u ≤ y/log y and z(1 + 1/z)z is increasing for z ≥ 1. Thus the above is ≥ (z′(1 + 1/z′)z′ )k ≥ (z′(1 + 1/z′)z′ )u−1 where z′ = y/(u log y). As z′ ≤ y/ √ k this is ≥ 1 y ( y u log y )u ( 1 + u log y y )y/log y = x y exp ( −u log log x + y log y log(1 + (log x )/y) ) . The stated inequality now follows on noting that log(1 + δ) ≥ δ/2 for 0 ≤ δ ≤ 1. □ When y is of the form y = (log x )a with a not too large, the upper bound of Theorem 7.6 and the lower bound of Theorem 7.8 are quite close, and we have Corollary 7.9If y = (log x )a and 1 ≤ a ≤ (log x )1/2 /(2 log log x ), then x 1−1/a exp ( log x 5a log log x ) <ψ (x ,y) < x 1−1/a exp ((log a + O (1)) log x a log log x ) . Proof The lower bound follows from Theorem 7.8 since log y ≤ (log x )/ (4a log log x ) in the range under consideration. As for the upper bound, we note that log u ≍ log log x , so that log log u = log log log x + O (1). Hence	{ "creation_date": "2025-02-03", "file_name": "(Cambridge Studies in Advanced Mathematics 97) Hugh L. Montgomery, Robert C. Vaughan - Multiplicative number theory I_ Classical theory-Cambridge University Press (2006).pdf", "file_path": "/home/suryaremanan/Documents/IMO/augmentoolkit/original/input/(Cambridge Studies in Advanced Mathematics 97) Hugh L. Montgomery, Robert C. Vaughan - Multiplicative number theory I_ Classical theory-Cambridge University Press (2006).pdf", "file_size": 2365282, "file_type": "application/pdf", "last_modified_date": "2025-02-03", "page_label": "209" }
210 Applications of the Prime Number Theorem log u + log log u = log log x − log a + O (1), and the result follows from Theorem 7.6. □ For 1 ≤ u ≤ 4 we may use the differential equation (7.4) and the initial condition (7.5) to derive formulæ for ρ(u) (see Exercise 7.1.6 below), but for larger u we take a different approach. Theorem 7.10 F or any real or complex number s we have ∫ ∞ 0 ρ(u)e−us du = exp ( C0 + ∫ s 0 e−z − 1 z dz ) (7.29) where C 0 is Euler’s constant. Conversely, for any u > 0 and any real σ0 we have ρ(u) = eC0 2πi ∫ σ0 +i ∞ σ0 −i ∞ exp (∫ s 0 e−z − 1 z dz ) eus ds . (7.30) Proof Let F (s) denote the integral on the left-hand side of (7.29); this is the Laplace transform of ρ(u). In view of the rapid decay of ρ(u) established in Lemma 7.1, we see that the integral converges for all s, and hence that F (s)i s an entire function. On integrating by parts we see that F (s) = 1 s + 1 s ∫ ∞ 1 ρ′(u)e−us du , and hence that (sF (s))′ =− ∫ ∞ 1 uρ′(u)e−us du . The differential–delay identity (7.4) for ρ(u) thus yields a differential equation for F (s), (sF (s))′ = e−s F (s). By separation of variables it follows that F (s) = F (0) exp (∫ s 0 e−z − 1 z dz ) . T o determine the value of F (0) we note that 1 = lim s→+∞ sF (s) = F (0) exp (∫ 1 0 e−z − 1 z dz + ∫ ∞ 1 e−z z dz ) . By integration by parts we see that ∫ 1 0 e−z − 1 z dz + ∫ ∞ 1 e−z z dz = ∫ ∞ 0 e−z log zd z = Ŵ′(1) =− C0 (7.31)	{ "creation_date": "2025-02-03", "file_name": "(Cambridge Studies in Advanced Mathematics 97) Hugh L. Montgomery, Robert C. Vaughan - Multiplicative number theory I_ Classical theory-Cambridge University Press (2006).pdf", "file_path": "/home/suryaremanan/Documents/IMO/augmentoolkit/original/input/(Cambridge Studies in Advanced Mathematics 97) Hugh L. Montgomery, Robert C. Vaughan - Multiplicative number theory I_ Classical theory-Cambridge University Press (2006).pdf", "file_size": 2365282, "file_type": "application/pdf", "last_modified_date": "2025-02-03", "page_label": "210" }
7.1 Numbers composed of small primes 211 by (C.12) and Theorem C.2. Hence F (0) = eC0 . An arithmetic proof of this is found in Exercise 7.1.7 below . Thus we have the identity (7.29), and (7.30) follows by applying the inverse Laplace transform to both sides.□ 7.1.1 Exercises 1. (Chowla & V ijayaraghavan 1947) Show that if f (x ) is a function that tends to inﬁnity in such a way that log f (x ) = o(log x ) then almost all integers n have a prime factor larger than f (n). That is lim x →∞ 1 x card{n ≤ x : P (n) > f (n)}= 1 where P (n) denotes the largest prime factor of n. 2. (de Bruijn 1951b) Let P (n) denote the largest prime factor of n. Show that ∑ n≤x log P (n) ∼ Dx log x where D = ∫∞ 0 ρ(u)(u + 1)−2 du is called Dickman’s constant . 3. (cf. Alladi & Erd ˝ os 1977) Let P (n) denote the largest prime factor of n. (a) Show that ∑ n≤x P (n) = ∑ √x <p≤x p [x p ] + O ( x 3/2 ) . (b) Show that the sum on the right above is = ∑ 1≤k≤√x k ∑ x /(k+1)<p≤x /k p + O ( x 3/2 ) . (c) Show that ∑ p≤y p = y2 2 log y + O ( y2 (log y)2 ) . (d) Show that ∞∑ k=1 k (1 k2 − 1 (k + 1)2 ) = π2 6 . (e) Conclude that ∑ n≤x P (n) = π2 12 x 2 log x + O ( x 2 (log x )2 ) . 4. Show that ρ(k) (u) has a jump discontinuity at u = k, and is continuous for u > k. 5. (a) Show that ρ(u) is convex upwards for all u ≥ 1. (b) Show that if u ≥ 2, then uρ(u) ≥ ρ(u − 1/2).	{ "creation_date": "2025-02-03", "file_name": "(Cambridge Studies in Advanced Mathematics 97) Hugh L. Montgomery, Robert C. Vaughan - Multiplicative number theory I_ Classical theory-Cambridge University Press (2006).pdf", "file_path": "/home/suryaremanan/Documents/IMO/augmentoolkit/original/input/(Cambridge Studies in Advanced Mathematics 97) Hugh L. Montgomery, Robert C. Vaughan - Multiplicative number theory I_ Classical theory-Cambridge University Press (2006).pdf", "file_size": 2365282, "file_type": "application/pdf", "last_modified_date": "2025-02-03", "page_label": "211" }
212 Applications of the Prime Number Theorem (c) Show that if u ≥ 2, then (2 u − 1)ρ(u) ≤ ρ(u − 1). 6. (a) Show that if 1 ≤ u ≤ 2, then ρ(u) = 1 − log u. (b) Show that if 2 ≤ u ≤ 3, then ρ(u) = 1 − log u + ∫ u 2 log(t − 1) t dt . (c) Show that if 3 ≤ u ≤ 4, then ρ(u) = 1 − log u + ∫ u 2 log(t − 1) t dt − ∫ u 3 (log u/t ) log( t − 2) t − 1 dt . 7. Let P (σ) = ∏ p≤y (1 − p−σ)−1 . (a) Explain why P (1) = ∑ p\|n⇒ p≤y 1 n = eC0 log y + O (1). (b) Show that if σ ≥ 1, then P ′ P (σ) ≪ log y. (c) Deduce that − P ′(1) = ∑ n p\|n⇒ p≤y log n n ≪ (log y)2 . (d) Conclude that ∑ n>x p\|n⇒ p≤y 1 n ≪ (log y)2 log x . (e) Show that ∑ n≤x p\|n⇒ p≤y 1 n = (log y) ∫ u 0 ψ( yv,y) yv d v + O (1) where u = (log x )/log y. (f) Deduce that ∫ ∞ 0 ρ(u) du = eC0 . (g) Show that ∑ ∞ n=1 nρ(n) = eC0 . 8. (Erd ˝ os & Nicolas 1981) Let α be ﬁxed, 0 <α< 1. (a) Let k be the least integer >α (log x )/log log x , put y = x 1/k , and set r = π( y). Show that there are at least (r k ) integers n ≤ x such that ω(n) >α (log x )/log log x . (b) Show that the number of integers n ≤ x such that ω(n) > α(log x )/log log x is at least x 1−α+o(1) .	{ "creation_date": "2025-02-03", "file_name": "(Cambridge Studies in Advanced Mathematics 97) Hugh L. Montgomery, Robert C. Vaughan - Multiplicative number theory I_ Classical theory-Cambridge University Press (2006).pdf", "file_path": "/home/suryaremanan/Documents/IMO/augmentoolkit/original/input/(Cambridge Studies in Advanced Mathematics 97) Hugh L. Montgomery, Robert C. Vaughan - Multiplicative number theory I_ Classical theory-Cambridge University Press (2006).pdf", "file_size": 2365282, "file_type": "application/pdf", "last_modified_date": "2025-02-03", "page_label": "212" }
7.1 Numbers composed of small primes 213 (c) Show that if σ> 1 and A ≥ 1, then the number of integers n ≤ x such that ω(n) >α (log x )/log log x is at most x σ A−k ∞∑ n=1 Aω(n) nσ . (d) Show that if A = log x and σ = 1 + (log log log x )/log log x , then the above is x 1−α+o(1) . 9. (de Bruijn 1966) Assume that 0 <σ ≤ 3/log y, and note that this interval covers a range that is not treated in Lemma 7.5. (a) Show that 1− p−σ ≍ σ log p, and hence deduce that ∏ p≤y (1 − p−σ)−1 ≤ exp (∑ p≤y log C σ log p ) ≤ exp (Cy log y log 4 σ log y ) (7.32) for a suitable constant C . (b) Write ∏ p≤y (1 − p−σ)−1 = (1 − y−σ)−π( y) ∏ p≤y 1 − y−σ 1 − p−σ = F1 · F2 , say . Show that F1 ≤ (1 − y−σ)−y/log y exp ( Cy (log y)2 log 4 σ log y ) . (c) Note that 1 − p−σ 1 − y−σ = 1 − ( y/p)σ − 1 yσ − 1 , (7.33) and hence deduce that the above is ≥ 1 − c log y/p log y , so that F2 ≤ exp ( C log y ∑ p≤y log y/p ) ≤ exp ( Cy /(log y)2 ) . (d) Conclude that ∏ p≤y (1 − p−σ)−1 ≤ (1 − y−σ)−y/log y exp ( Cy (log y)2 log 4 σ log y ) for 0 <σ ≤ 3/log y. 10. (de Bruijn 1966) Lemma 7.5 suffers from a loss of precision when 3/log y ≤ σ ≤ (log log y)/log y. T o obtain a reﬁned estimate in this range, write ∏ p≤y (1 − p−σ)−1 = F1 · F2 · F3	{ "creation_date": "2025-02-03", "file_name": "(Cambridge Studies in Advanced Mathematics 97) Hugh L. Montgomery, Robert C. Vaughan - Multiplicative number theory I_ Classical theory-Cambridge University Press (2006).pdf", "file_path": "/home/suryaremanan/Documents/IMO/augmentoolkit/original/input/(Cambridge Studies in Advanced Mathematics 97) Hugh L. Montgomery, Robert C. Vaughan - Multiplicative number theory I_ Classical theory-Cambridge University Press (2006).pdf", "file_size": 2365282, "file_type": "application/pdf", "last_modified_date": "2025-02-03", "page_label": "213" }
214 Applications of the Prime Number Theorem where the Fi are products over the intervals p ≤ exp(1/σ), exp(1 /σ) < p ≤ y/exp(1/σ), and y/exp(1/σ) < p ≤ y, respectively . (a) Use (7.32) to show that F1 ≤ exp ( C σe1/σ) . (b) Use Lemma 7.5 to show that F2 ≤ exp (Cy 1−σ e1/σ log y ) . (c) Use the identity (7.33) to show that 1 − p−σ 1 − y−σ ≥ 1 − cσ log y/p yσ , and hence deduce that F3 ≤ (1 − y−σ)−π( y) exp ( C σ ∑ p≤y log y/p yσ ) ≤ (1 − y−σ)−y/log y exp (y1−σ (log y)2 + C σy1−σ log y ) . (d) Conclude that ∏ p≤y (1 − p−σ)−1 ≤ (1 − y−σ)−y/log y exp (C σy1−σ log y ) when 3 /log y ≤ σ ≤ (log log y)/log y. 11. (de Bruijn 1966) (a) For σ> 0 let f (σ) = x σ(1 − y−σ)−y/log y . Show that f (σ) is mini- mized precisely when σ = log(1 + y/log x ) log y . (b) Show that for the above σ, f (σ) = exp (log x log y log (y + log x log x ) + y log y log (y + log x y )) . (c) Show that if y ≤ log x , then ψ(x ,y) ≤ exp (log x log y log (y + log x log x ) + y log y ( 1 + O (1 log y )) log (y + log x y )) . (d) Show that if log x ≤ y ≤ (log x )2 , then ψ(x ,y) ≤ exp (log x log y ( 1 + O (1 log y )) log (y + log x log x ) + y log y log (y + log x y )) .	{ "creation_date": "2025-02-03", "file_name": "(Cambridge Studies in Advanced Mathematics 97) Hugh L. Montgomery, Robert C. Vaughan - Multiplicative number theory I_ Classical theory-Cambridge University Press (2006).pdf", "file_path": "/home/suryaremanan/Documents/IMO/augmentoolkit/original/input/(Cambridge Studies in Advanced Mathematics 97) Hugh L. Montgomery, Robert C. Vaughan - Multiplicative number theory I_ Classical theory-Cambridge University Press (2006).pdf", "file_size": 2365282, "file_type": "application/pdf", "last_modified_date": "2025-02-03", "page_label": "214" }
7.2 Numbers composed of large primes 215 12. (Erd ˝ os 1963) Show that ψ(x ,log x ) = exp ( (2 log 2 + o(1)) log x log log x ) . 13. (de Bruijn 1966) Show that if a is ﬁxed, 0 < a < 1, then ψ(x ,(log x )a ) = exp((1/a − 1 + o(1))(log x )a ). 14. Let ψ2 (x ,y) denote the number of square-free integers n ≤ x composed entirely of primes p ≤ y. (a) Show that ψ2 (x ,y) = ∑ d ≤x p\|d ⇒ p≤y µ(d )ψ(x /d 2 ,y). (b) (Ivi´ c) Let δ> 0 be ﬁxed. Then ψ2 (x ,y) ∼ 6 π2 ψ(x ,y) uniformly for x δ ≤ y ≤ x . (c) Show that ψ2 (x ,log x ) = ψ(x ,log x )1/2+o(1) . (d) Show that if a > 1 and y ≥ (log x )a , then ψ2 (x ,y) = ψ(x ,y)1+o(1) . (e) Show that if 0 < a < 1 and y ≤ (log x )a , then ψ2 (x ,y) = ψ(x ,y)o(1) . (f) Show that ψ2 (x ,c log x ) = ψ(x ,c log x )φ(c)+o(1) for any ﬁxed c > 0, where φ(c) = ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ c log 2 (c + 1) log( c + 1) − c log c (0 < c ≤ 2), c log c − (c − 1) log( c − 1) (c + 1) log( c + 1) − c log c (c ≥ 2). 7.2 Numbers composed of large primes Let /Phi1 (x ,y) denote the number of integers n ≤ x composed entirely of primes p ≥ y. The number 1 is such a number as it is an empty product. Thus it is clear that if y > x , then /Phi1 (x ,y) = 1 (7.34) Also, if x 1/2 ≤ y ≤ x , then /Phi1 (x ,y) = π(x ) − π( y−) + O (1) = x log x − y log y + O ( x (log x )2 ) (7.35) For smaller values of y we show that /Phi1 (x ,y) ∼ w(u)x log y (7.36)	{ "creation_date": "2025-02-03", "file_name": "(Cambridge Studies in Advanced Mathematics 97) Hugh L. Montgomery, Robert C. Vaughan - Multiplicative number theory I_ Classical theory-Cambridge University Press (2006).pdf", "file_path": "/home/suryaremanan/Documents/IMO/augmentoolkit/original/input/(Cambridge Studies in Advanced Mathematics 97) Hugh L. Montgomery, Robert C. Vaughan - Multiplicative number theory I_ Classical theory-Cambridge University Press (2006).pdf", "file_size": 2365282, "file_type": "application/pdf", "last_modified_date": "2025-02-03", "page_label": "215" }
216 Applications of the Prime Number Theorem 0 1 1 Figure 7.2 Buchstab’s function w(u) and its horizontal asymptote e−C0 for 1 ≤ u ≤ 4. where u = (log x )/log y and w(u) is a function determined by the initial con- dition w(u) = 1/u (7.37) for 1 < u ≤ 2 and for u > 2 by the differential–delay equation (uw(u))′ = w(u − 1). (7.38) Before proceeding further we ﬁrst derive some of the simplest properties of the function w(u) depicted in Figure 7.2. By integrating (7.38) we deduce that uw(u) = ∫u−1 1 w(v) d v + C for u > 2, and by letting u tend to 2 we ﬁnd that C = 1 so that uw(u) = ∫ u−1 1 w(v) d v + 1 (7.39) for u ≥ 2. From this it is evident that if w(v) ≤ 1 for v ≤ u − 1, then w(v) ≤ 1 for v ≤ u, and that if w(v) ≥ 1/2 for v ≤ u − 1, then w(v) ≥ 1/2 for v ≤ u. Thus we conclude that 1 /2 ≤ w(u) ≤ 1 for all u > 1. From the identity uw′(u) = w(u − 1) − w(u) we deduce that \|w′(u)\|≤ 1/(2u) for all u > 2. Let M (u) = maxv≥u \|w′(v)\|. Since w(u − 1) − w(u) =− w′(ξ) for some ξ, u − 1 <ξ< u, we know that M (u) ≤ M (u − 1)/u. Let k be chosen so that 1 < u − k ≤ 2. By using the above inequality k times we ﬁnd that M (u) ≤ M (u − k) u(u − 1) ··· (u − k + 1) ≪ 1 Ŵ(u + 1) . That is, w′(u) ≪ 1 Ŵ(u + 1) (7.40)	{ "creation_date": "2025-02-03", "file_name": "(Cambridge Studies in Advanced Mathematics 97) Hugh L. Montgomery, Robert C. Vaughan - Multiplicative number theory I_ Classical theory-Cambridge University Press (2006).pdf", "file_path": "/home/suryaremanan/Documents/IMO/augmentoolkit/original/input/(Cambridge Studies in Advanced Mathematics 97) Hugh L. Montgomery, Robert C. Vaughan - Multiplicative number theory I_ Classical theory-Cambridge University Press (2006).pdf", "file_size": 2365282, "file_type": "application/pdf", "last_modified_date": "2025-02-03", "page_label": "216" }
7.2 Numbers composed of large primes 217 for u > 2. Since w′(u) tends to 0 rapidly , it follows that the integral ∫∞ 2 w′(v) d v converges absolutely , and hence we see that lim u→∞ w(u) exists. Since it is to be expected that /Phi1 (x ,y) is approximately x ∏ p<y (1 − 1/p) when y is small, it is not surprising that lim u→∞ w(u) = e−C0 . (7.41) W e shall prove this later, as a consequence of Theorem 7.12. First we establish the basic asymptotic estimate (7.36). Theorem 7.11(Buchstab) Let /Phi1 (x ,y) denote the number of positive integers n ≤ x composed entirely of prime numbers p ≥ y, and let w(u) be deﬁned as above. Then /Phi1 (x ,y) = w(u)x log y − y log y + O ( x (log x )2 ) (7.42) uniformly for 1 ≤ u ≤ U and all y ≥ 2. Here u = (log x )/log y, which is to say that y = x 1/u . The term −y/log y can be included in the error term when y ≪ x /log x but, in view of (7.35), has to be present when y is close to x . It might be difﬁcult to prove that the above holds uniformly for all u ≥ 1 because of the precise form of the error term, but the weaker assertion (7.36) can be shown to hold for u≥ 1 + ε, since sieve methods can be used when u is large. Proof The number of positive integers n ≤ x whose least prime factor is p is exactly /Phi1 (x /p, p). Hence by classifying integers according to their least prime factor we see that /Phi1 (x ,y) = 1 + ∑ y≤ p≤x /Phi1 (x /p, p). (7.43) This is an identity of Buchstab; similar ‘Buchstab identities’ are important in sieve theory . W e show by induction onU that /Phi1 (x ,y) = w(u)x log y − y log y + O ( x (log x )2 ) (7.44) for U ≤ u ≤ U + 1. When U = 1 this is (7.35), and it is only in this ﬁrst range that the second main term is signiﬁcant. For the inductive step we apply (7.43) withy = x 1/u and with y = x 1/U and subtract to see that /Phi1 ( x ,x 1/u ) = /Phi1 ( x ,x 1/U ) + ∑ x 1/u ≤ p<x 1/U /Phi1 (x /p, p).	{ "creation_date": "2025-02-03", "file_name": "(Cambridge Studies in Advanced Mathematics 97) Hugh L. Montgomery, Robert C. Vaughan - Multiplicative number theory I_ Classical theory-Cambridge University Press (2006).pdf", "file_path": "/home/suryaremanan/Documents/IMO/augmentoolkit/original/input/(Cambridge Studies in Advanced Mathematics 97) Hugh L. Montgomery, Robert C. Vaughan - Multiplicative number theory I_ Classical theory-Cambridge University Press (2006).pdf", "file_size": 2365282, "file_type": "application/pdf", "last_modified_date": "2025-02-03", "page_label": "217" }
218 Applications of the Prime Number Theorem Choose u p so that p = (x /p)1/u p . Then the above is /Phi1 ( x ,x 1/U ) + ∑ x 1/u ≤ p<x 1/U /Phi1 ( x /p,(x /p)1/u p ) . But u p = (log x )/log p − 1 ∈ [U − 1,U ], so by the inductive hypothesis, when U ≥ 2, the above is U w(U )x log x + O ( x (log x )2 ) + ∑ x 1/u ≤ p<x 1/U (u p w(u p )x p log x /p + O ( x p(log x )2 ) + O ( p log p )) . The sum over p of the ﬁrst error term is ≪ x /(log x )2 , and the sum over p of the second is ≪ x 2/U /(log x )2 , which is acceptable since U ≥ 2. T o estimate the contribution of the main term in the sum we write the Prime Number Theorem in the formπ(t ) = li(t ) + R(t ), apply Riemann–Stieltjes integration, and integrate the term involving R(t ) by parts, to see that the sum of the main term is ∫ x 1/U x 1/u x w (log x log t − 1 ) t (log t )2 dt + [ f (t ) R(t ) ⏐ ⏐ ⏐ x 1/U − x 1/u − − ∫ x 1/U x 1/u R(t ) df (t ) (7.45) where f (t ) = x w (log x log t − 1 ) t log t . Since f ′(t ) ≪ x /(t 2 log t ) and R(t ) ≪ t /(log t ) A , the terms involving R(t ) contribute an amount ≪U x /(log x ) A . By the change of variables v = (log x )/log t − 1 we see that the ﬁrst integral in (7.45) is x log x ∫ u−1 U −1 w(v) d v, which by (7.39) is = x log x (uw(u) − U w(U )). On combining our estimates we obtain (7.44), so the inductive step is complete. □ W e now derive formulæ for w(u) similar to those in Theorem 7.10 involving ρ(u). Theorem 7.12 If ℜs > 0, then s + s ∫ ∞ 1 w(u)e−us du = exp ( −C0 + ∫ s 0 1 − e−z z dz ) (7.46)	{ "creation_date": "2025-02-03", "file_name": "(Cambridge Studies in Advanced Mathematics 97) Hugh L. Montgomery, Robert C. Vaughan - Multiplicative number theory I_ Classical theory-Cambridge University Press (2006).pdf", "file_path": "/home/suryaremanan/Documents/IMO/augmentoolkit/original/input/(Cambridge Studies in Advanced Mathematics 97) Hugh L. Montgomery, Robert C. Vaughan - Multiplicative number theory I_ Classical theory-Cambridge University Press (2006).pdf", "file_size": 2365282, "file_type": "application/pdf", "last_modified_date": "2025-02-03", "page_label": "218" }
7.2 Numbers composed of large primes 219 where C 0 is Euler’s constant. If u > 1 and σ0 > 0, then w(u) = 1 2πi ∫ σ0 +i ∞ σ0 −i ∞ ( exp (∫ ∞ s e−z z dz ) − 1 ) eus ds . (7.47) Since the right-hand side of (7.46) is an entire function, we see that the Laplace transform of w(u) is entire apart from a simple pole at s = 0 with residue e−C0 . Proof Let G (s) denote the left-hand side of (7.46). Then(G (s) s )′ =− ∫ ∞ 1 w(u)ue −us du . By integrating by parts we see that this is [w(u)ue −us s ⏐ ⏐ ⏐ ∞ 1 − 1 s ∫ ∞ 2 w(u − 1)e−us du = −e−s G (s) s2 by (7.37) and (7.38). That is, G ′(s) = G (s) 1 − e−s s , which by the method of separation of variables implies that G (s) = A exp (∫ s 0 1 − e−z z dz ) where A is a positive constant. T o determine the value of A we note that 1 = lim s→∞ G (s) s = A exp (∫ 1 0 1 − e−z z dz − ∫ ∞ 1 e−z z dz ) . From (7.31) we deduce that A = e−C0 , and hence we have (7.46). T o obtain (7.47) it sufﬁces to take the inverse Laplace transform, since ∫ s 0 1 − e−z z dz = ∫ ∞ s e−z z dz + log s + C0 . □ 7.2.1 Exercises 1. By using (7.31), or otherwise, show that ∫ s 0 1 − e−z z dz = C0 + log s + ∫ ∞ s e−z z dz when ℜs > 0. 2. (a) Show that w(u) = 1 + log(u − 1) u for 2 ≤ u ≤ 3.	{ "creation_date": "2025-02-03", "file_name": "(Cambridge Studies in Advanced Mathematics 97) Hugh L. Montgomery, Robert C. Vaughan - Multiplicative number theory I_ Classical theory-Cambridge University Press (2006).pdf", "file_path": "/home/suryaremanan/Documents/IMO/augmentoolkit/original/input/(Cambridge Studies in Advanced Mathematics 97) Hugh L. Montgomery, Robert C. Vaughan - Multiplicative number theory I_ Classical theory-Cambridge University Press (2006).pdf", "file_size": 2365282, "file_type": "application/pdf", "last_modified_date": "2025-02-03", "page_label": "219" }
220 Applications of the Prime Number Theorem (b) Show that w(u) = 1 u ( 1 + log(u − 1) + ∫ u 3 log(v − 2) v − 1 d v ) for 3 ≤ u ≤ 4. (c) Show that w(u) = 1 u ( 1 + log(u − 1) + ∫ u 3 log(v − 2) v − 1 d v + ∫ u 4 log u−1 v−1 log(v − 3) v − 2 d v ) for 4 ≤ u ≤ 5. 3. (Friedlander 1972) Let S be a set of positive integers not exceeding X , and suppose that ( a,b) ≤ Y whenever a ∈ S, b ∈ S, a ̸=b. Let M ( X ,Y ) denote the maximum cardinality of all such sets S. (a) Let S0 be the set of those positive integers n ≤ X such that if d \|n, d < n, then d ≤ Y . Show that card S0 = M ( X ,Y ). (b) Show that if Y ≤ X 1/2 , then M ( X ,Y ) = 1 + π( X ) − π(Y ) + ∑ p≤Y /Phi1 (Y, p). (c) Show that if X 1/2 < Y ≤ X , then M ( X ,Y ) = 1 + π( X ) − π(Y ) + ∑ p<X /Y /Phi1 (Y, p) + ∑ X /Y ≤ p≤Y /Phi1 ( X /p, p). 7.3 Primes in short intervals Let Jacobsthal’s function g(q ) be the length of the longest gap between con- secutive reduced residues modulo q . W e show that there are long gaps between primes by showing that there exist integers q for which g(q ) is large. Since the average gap between consecutive reduced residues (mod q )i s q /ϕ(q ), it is obvious that g(q ) ≥ q ϕ(q ) . If p1 < p2 < ··· < pk are the distinct primes dividing q , then by the Chinese Remainder Theorem there is an x such that x ≡− i (mod pi ) for 1 ≤ i ≤ k. Then ( x + i,q ) > 1 for 1 ≤ i ≤ k, and hence g(q ) ≥ ω(q ) + 1.	{ "creation_date": "2025-02-03", "file_name": "(Cambridge Studies in Advanced Mathematics 97) Hugh L. Montgomery, Robert C. Vaughan - Multiplicative number theory I_ Classical theory-Cambridge University Press (2006).pdf", "file_path": "/home/suryaremanan/Documents/IMO/augmentoolkit/original/input/(Cambridge Studies in Advanced Mathematics 97) Hugh L. Montgomery, Robert C. Vaughan - Multiplicative number theory I_ Classical theory-Cambridge University Press (2006).pdf", "file_size": 2365282, "file_type": "application/pdf", "last_modified_date": "2025-02-03", "page_label": "220" }
7.3 Primes in short intervals 221 These observations can be combined: It can be shown that g(q ) ≫ q ω(q ) ϕ(q ) . (7.48) This is not quite enough to produce long gaps between primes, but for certain q we improve on the above to establish Lemma 7.13 Let P = P (z) = ∏ p≤z p. Then lim z→∞ g ( P (z) ) z =∞ . This immediately yields Theorem 7.14 (W estzynthius) Let p n denote the n th prime number . Then lim sup n→∞ pn+1 − pn log pn =∞ . Proof of Theorem 7.14 Suppose that N = g( P ) − 1 and that M is chosen, P ≤ M < 2 P , so that ( M + m,P ) > 1 for 1 ≤ m ≤ N . But M + m > P ≥ ( M + m,P ), and hence M + m is composite because it has the proper divisor ( M + m,P ). If n is chosen so that pn is the largest prime not exceeding M , then pn+1 − pn ≥ g( P ) and pn < 2 P , which is < e2z when z is large. Hence pn+1 − pn log pn ≥ g( P ) 2z which tends to inﬁnity as z →∞ . □ Proof of Lemma 7.13 Let L be large and ﬁxed, and put N = [zL /3]. W e show that if z > z0 (L ), then there exists an integer M such that ( M + n,P (z)) > 1 for 1 ≤ n ≤ N . Put P1 = ∏ p≤L p, P2 = ∏ L <p≤L L p, P3 = ∏ L L <p≤z/3 p, P4 = ∏ z/3<p≤z p, and let N be the set of those integers n,1 ≤ n ≤ N , such that ( n,P1 P3 ) = 1. The members of N are (i) 1; (ii) integers n composed entirely of prime factors of P2 ; (iii) primes p, z/3 < p ≤ N . Thus card N ≤ 1 + ψ( N ,L L ) + π( N ) − π(z/3). If z is sufﬁciently large, then L L < log N , so that ψ( N ,L L ) < N ε by Corol- lary 7.9. Hence card N <π ( N ).	{ "creation_date": "2025-02-03", "file_name": "(Cambridge Studies in Advanced Mathematics 97) Hugh L. Montgomery, Robert C. Vaughan - Multiplicative number theory I_ Classical theory-Cambridge University Press (2006).pdf", "file_path": "/home/suryaremanan/Documents/IMO/augmentoolkit/original/input/(Cambridge Studies in Advanced Mathematics 97) Hugh L. Montgomery, Robert C. Vaughan - Multiplicative number theory I_ Classical theory-Cambridge University Press (2006).pdf", "file_size": 2365282, "file_type": "application/pdf", "last_modified_date": "2025-02-03", "page_label": "221" }
222 Applications of the Prime Number Theorem W e choose M ≡ 0 (mod P1 P3 ), so that ( M + n,P1 P3 ) > 1i f1 ≤ n ≤ N , n /∈ N. T o bound the number of n ∈ N such that ( M + n,P2 ) = 1 we average as in the proof of Lemma 3.5. Clearly q∑ m=1 ∑ n∈N (m+n,q )=1 1 = ∑ n∈N q∑ m=1 (m+n,q )=1 1 = ∑ n∈N ϕ(q ) = ϕ(q ) card N for any integer q . Hence min m ∑ n∈N (m+n,q )=1 1 ≤ (card N) ∏ p\|q ( 1 − 1 p ) . By taking q = P2 we see that there is an M (mod P2 ) such that card{n ∈ N :( M + n,P2 ) = 1}≤ (card N) ∏ p\| P2 ( 1 − 1 p ) . For such an M , card{1 ≤ n ≤ N :( M + n,P1 P2 P3 ) = 1}≤ π( N ) ∏ p\| P2 ( 1 − 1 p ) . By Mertens’ theorem (Theorem 2.7(e)), the product on the right is ∼ 1/L as L →∞ . Suppose that L is chosen sufﬁciently large to ensure that this product is ≤ 3/(2L ). Then the right-hand side above is ≲ 3 N 2L log N ∼ z 2 log z . The number of primes dividing P4 is π(z) − π(z/3) ∼ 2z/(3 log z)a s z →∞ . Thus if z is large, then there are more such primes than there are integers n, 1 ≤ n ≤ N , for which ( M + n,P1 P2 P3 ) = 1. Hence for each such n we may as- sociate a prime pn , pn \| P4 , in a one-to-one manner, and take M ≡− n (mod pn ). Then ( M + n,P4 ) > 1 and we are done. □ The success of the argument just completed can be attributed to the fact that the number of n,1 ≤ n ≤ N , for which ( n,P1 P3 ) = 1 is considerably smaller than N ∏ p\| P1 P3 (1 − 1/p). By considering how L may be chosen as a function of z we obtain a quantitative improvement of Lemma 7.13 and hence also of Theorem 7.14. Theorem 7.15(Rankin) Let p n denote the n th prime number in increasing order . There is a constant c > 0 such that lim sup n→∞ pn+1 − pn ( (log pn )(log log pn )(log log log log pn ) (log log log pn )2 )≥ c.	{ "creation_date": "2025-02-03", "file_name": "(Cambridge Studies in Advanced Mathematics 97) Hugh L. Montgomery, Robert C. Vaughan - Multiplicative number theory I_ Classical theory-Cambridge University Press (2006).pdf", "file_path": "/home/suryaremanan/Documents/IMO/augmentoolkit/original/input/(Cambridge Studies in Advanced Mathematics 97) Hugh L. Montgomery, Robert C. Vaughan - Multiplicative number theory I_ Classical theory-Cambridge University Press (2006).pdf", "file_size": 2365282, "file_type": "application/pdf", "last_modified_date": "2025-02-03", "page_label": "222" }
7.3 Primes in short intervals 223 Proof W e repeat the argument in the proof of Lemma 7.13, with the sole change that L is allowed to depend on z.I f L is chosen so that ψ( N ,L L ) < N (log N )2 , (7.49) then L = o(log N ), and hence ψ( N ,L L ) = o ( z log N ) . Since z/log N ≤ z/log z ≪ π(z/3), it follows that ψ( N ,L L ) = o(π(z/3)), and the proof proceeds as before. By Theorem 7.6 we see that ψ ( N ,N 1/u ) < N (log N )2 if u log u ≥ 3 log log N , which is the case if u ≥ 4(log log N )/log log log N . T aking u = (log N )/log L L , we deduce that (7.49) holds if L log L < (log N )(log log log N ) 4 log log N . This is satisﬁed if L < (log N )(log log log N ) 4(log log N )2 , since then log L < log log N . Since N > z when L ≥ 3, we conclude that we may take L = (log z)(log log log z) 4(log log z)2 . Hence g ( P (z) ) > z(log z)(log log log z) 13(log log z)2 for all z > z0 , and this gives the stated result. □ Concerning the maximum number of primes in a short interval, by the Brun– Titchmarsh inequality (Theorem 3.9) and the Prime Number Theorem we see that π(x + y) − π(x ) < (2 + ε)π( y) for y > y0 (ε). Let ρ( y) = lim sup x →∞ (π(x + y) − π(x )). (7.50)	{ "creation_date": "2025-02-03", "file_name": "(Cambridge Studies in Advanced Mathematics 97) Hugh L. Montgomery, Robert C. Vaughan - Multiplicative number theory I_ Classical theory-Cambridge University Press (2006).pdf", "file_path": "/home/suryaremanan/Documents/IMO/augmentoolkit/original/input/(Cambridge Studies in Advanced Mathematics 97) Hugh L. Montgomery, Robert C. Vaughan - Multiplicative number theory I_ Classical theory-Cambridge University Press (2006).pdf", "file_size": 2365282, "file_type": "application/pdf", "last_modified_date": "2025-02-03", "page_label": "223" }
224 Applications of the Prime Number Theorem Thus ρ( y) < (2 + ε)π( y). V ery little is known about ρ( y). It was once conjec- tured that π( M + N ) ≤ π( M ) + π( N ) (7.51) for M > 1,N > 1, but there is now serious doubt as to the validity of this inequality . Indeed, it seems likely that ρ( y) >π ( y) for all large y. T o see why , let ρ( N ) = max M M +N∑ n=M +1 p\|n⇒ p>N 1. (7.52) Clearly ρ( N ) ≤ ρ( N ). W e expect that ρ( N ) = ρ( N ) (7.53) for all N , since this would follow from the Prime k-tuple conjecture. Let a 1 ,a2 ,..., ak , be given integers. Then there exist inﬁnitely many positive integers n such that n + a1 ,n + a2 ,..., n + ak are all prime, provided that for every prime number p there is an integer n such that(n + ai , p) = 1 for i = 1,2,..., k. W e now show that ρ( N ) >π ( N ) for all large N , so that (7.51) and (7.53) are inconsistent. Theorem 7.16There is an absolute constant N 0 such that if N > N0 then ρ( N ) − π( N ) ≫ N (log N )−2 . Proof Suppose that N is even and that N > 2. Then for every M , M +N∑ n=M +1 p\|n⇒ p>N 1 = M +N∑ n=M +1 p\|n⇒ p≥N 1 ≥ M +N −1∑ n=M +1 p\|n⇒ p>N −1 1. Hence ρ( N ) ≥ ρ( N − 1) when N is even, N > 2, so it sufﬁces to treat the case when N is odd, say N = 2K + 1. Let P(K ) denote the set of integers n with K /(2 log K ) < \|n\|≤ K and \|n\| prime. Then card P(K ) = 2(π(K ) − π(K /(2 log K ))), so by Theorem 6.9, card P(K ) = π(2K + 1) + (c + o(1)) K (log K )2 where c = 2 log 2 − 1 > 0. W e now show that P(K ) can be translated to form a set of integers {M + n : n ∈ P(K )} with each member coprime to ∏ p≤N p. By the Chinese Remainder Theorem it sufﬁces to show that for every prime	{ "creation_date": "2025-02-03", "file_name": "(Cambridge Studies in Advanced Mathematics 97) Hugh L. Montgomery, Robert C. Vaughan - Multiplicative number theory I_ Classical theory-Cambridge University Press (2006).pdf", "file_path": "/home/suryaremanan/Documents/IMO/augmentoolkit/original/input/(Cambridge Studies in Advanced Mathematics 97) Hugh L. Montgomery, Robert C. Vaughan - Multiplicative number theory I_ Classical theory-Cambridge University Press (2006).pdf", "file_size": 2365282, "file_type": "application/pdf", "last_modified_date": "2025-02-03", "page_label": "224" }
7.3 Primes in short intervals 225 number p ≤ N there is a residue class r p (mod p) that contains no element of P(K ). Obviously each element of P(K ) is coprime to each prime p ≤ K /(2 log K ), so we may take r p = 0 for such primes. It remains to treat the primes p for which K /(2 log K ) < p ≤ 2K + 1. This is accomplished by means of a clever application of Lemma 7.13. Suppose that K /(2 log K ) < p ≤ 2K + 1. W e show that there is an r p such that if \|hp + r p \|≤ K , then hp + r p /∈ P(K ). By Lemma 7.13 there is an interval J = [ M1 − 3 log K ,M1 + 3 log K ] in which every integer j is divisible by a prime p j with p j ≤ 1 3 log K . By the Chinese Remainder Theorem, we can choose r p so that r p ≡ M1 p (mod p j ) for each j ∈ J . This can be done with 0 <r p ≤ exp ( ϑ( 1 3 log K ) ) < K 1/2 .I f \|h\|≤ 3 log K then h = j − M1 for some j ∈ J and so h ≡− M1 (mod p j ). Hence hp + r p ≡− M1 p + r p ≡ 0 (mod p j ), which implies that hp + r p /∈ P(K ). On the other hand, if \|h\| > 3 log K , then \|hp + r p \|≥ (3 2 − o(1) ) K > K , so that hp + r p /∈ P(K ) in this case also. Since the arithmetic progression hp + r p has no element in common with P(K ) the proof is complete. □ 7.3.1 Exercises 1. Show that the function ρ( N ) is weakly increasing. 2. (a) Show that in the prime k-tuple conjecture, the hypothesis that for every prime p the numbers a j do not cover all residue classes (mod p)i s satisﬁed for all p > k, so that it is enough to verify the hypothesis for p ≤ k (a ﬁnite calculation for any given set of a j ). (b) Prove the converse of the prime k-tuple conjecture: If there exist in- ﬁnitely many integers n for which n + a j is prime for all j ,1 ≤ j ≤ k, then for every prime p there is a residue class x (mod p) such that x + a j ̸≡0 (mod p)(1 ≤ j ≤ k). 3. Show that g(q ) ≫ q ω(q )/ϕ(q ). 4. (cf. Erd ˝ os 1951) Show that if 0 < c < 1/2 then there exist arbitrarily large numbers x such that the interval ( x ,x + c(log x )/log log x ) contains no square-free number. 5. (cf. Erd ˝ os 1946, Montgomery 1987) Suppose that 2 ≤ h ≤ x . Let P de- note the set of all primes p ≤ h, let D denote the set of positive integers composed entirely of primes in P, and let f (n) = ∏ p\|n,p∈P(1 − 1/p). (a) Show that f (n) = ∑ d \|n,d ∈D µ(d )/d . (b) Show that ∑ x <n≤x +h f (n) = 6 π2 h + O (log h) uniformly in x .	{ "creation_date": "2025-02-03", "file_name": "(Cambridge Studies in Advanced Mathematics 97) Hugh L. Montgomery, Robert C. Vaughan - Multiplicative number theory I_ Classical theory-Cambridge University Press (2006).pdf", "file_path": "/home/suryaremanan/Documents/IMO/augmentoolkit/original/input/(Cambridge Studies in Advanced Mathematics 97) Hugh L. Montgomery, Robert C. Vaughan - Multiplicative number theory I_ Classical theory-Cambridge University Press (2006).pdf", "file_size": 2365282, "file_type": "application/pdf", "last_modified_date": "2025-02-03", "page_label": "225" }
226 Applications of the Prime Number Theorem (c) Show that ϕ(n) n ≥ f (n) − ∑ p\|n p>h 1 p . (d) Among those primes p > h that divide an integer in the interval ( x ,x + h], let Q be those for which p ≤ h log x , and R those for which p > h log x . Show that ∑ p∈Q 1 p ≪ log log log x . (e) Explain why ∏ p∈R U <p≤2U p ⏐ ⏐ ⏐ ⏐ ∏ x <n≤x +h n, and deduce that card{ p ∈ R : U < p ≤ 2U }≪ h log x log U . (f) By summing over U = 2k h log x , show that ∑ p∈R 1 p ≪ 1 log(h log x ) . (g) Show that 6 π2 h + O (log h) + O (log log log x ) ≤ ∑ x <n≤x +h ϕ(n) n ≤ 6 π2 h + O (log h). 6. (cf. Pillai & Chowla 1930) Show that there is an absolute constant c > 0 such that there exist arbitrarily large x for which ϕ(n)/n < 1/4 when x < n ≤ x + c log log log x . Deduce that ∑ n≤x ϕ(n) n − 6 π2 x = /Omega1 (log log log x ). 7. (Hausman & Shapiro 1973; cf. Montgomery & V aughan 1986) (a) Show that q∑ n=1 ⎛ ⎜ ⎝ h∑ m=1 (m+n,q )=1 1 − ϕ(q ) q h ⎞ ⎟ ⎠ 2 = ϕ(q )2 q ∑ r \|q r >1 µ(r )2 r 2 ϕ(r )2 {h/r }(1 −{ h/r }) ∏ p\|q p∤r p( p − 2) ( p − 1)2 .	{ "creation_date": "2025-02-03", "file_name": "(Cambridge Studies in Advanced Mathematics 97) Hugh L. Montgomery, Robert C. Vaughan - Multiplicative number theory I_ Classical theory-Cambridge University Press (2006).pdf", "file_path": "/home/suryaremanan/Documents/IMO/augmentoolkit/original/input/(Cambridge Studies in Advanced Mathematics 97) Hugh L. Montgomery, Robert C. Vaughan - Multiplicative number theory I_ Classical theory-Cambridge University Press (2006).pdf", "file_size": 2365282, "file_type": "application/pdf", "last_modified_date": "2025-02-03", "page_label": "226" }
7.3 Primes in short intervals 227 (b) Use the inequality {α}(1 −{ α}) ≤ α to show that q∑ n=1 ⎛ ⎜ ⎝ h∑ m=1 (m+n,q )=1 1 − ϕ(q ) q h ⎞ ⎟ ⎠ 2 ≤ hϕ(q ). 8. (Erd ˝ os 1951) (a) For a positive integer q , let S(q ) denote the set of those residue classes s modulo q 2 such that ( s,q ) is a perfect square. Show that if q is square-free, then S(q ) contains exactly ∏ p\|q ( p2 − p + 1) elements. (b) Show that if q is square-free and 1 ≤ h ≤ q 2 , then there is an integer a such that the number of members of S(q ) in the interval ( a,a + h] is at most h ∏ p\|q ( 1 − 1 p + 1 p2 ) . (c) From now on, suppose that q is the product of those primes p ≤ y such that p ≡ 3 (mod 4). By recalling Corollary 4.12, or otherwise, show that the expression above is ≍ h/√ log y. (d) Show that if an integer n can be expressed as a sum of two squares, then n ∈ S(q ). (e) Let R be the set of those primes p, y < p ≤ Cy , such that p ≡ 3 (mod 4). Here C is an absolute constant, taken to be sufﬁciently large to ensure that R has at least y/log y elements. Note that such a constant exists, in view of Exercise 4.3.5(e). Let r denote the product of all members of R. Suppose that the number of members of S(q ) lying in the interval ( a,a + h]i s < y/log y. For each s ∈ S(q ) satisfying a < s ≤ a + h, associate a prime p ∈ R. Suppose that the integer b is chosen modulo p2 so that s + bq 2 ≡ p (mod p2 ). Show that the interval (a + bq 2 ,a + bq 2 + h] does not contain a sum of two squares. (f) Show that a and b can be chosen so that 0 < a + bq 2 < (qr )2 . (g) Show that log qr ≪ y. (h) Show that this construction succeeds with h ≍ y/√ log y ≫ (log qr )/(log log qr )1/2 . (i) Conclude that there exist arbitrarily large x such that there is no sum of two squares between x and x + c(log x )/(log log x )1/2 . Here c is a suit- ably small positive constant. (Note that a stronger result is established in the next exercise.)	{ "creation_date": "2025-02-03", "file_name": "(Cambridge Studies in Advanced Mathematics 97) Hugh L. Montgomery, Robert C. Vaughan - Multiplicative number theory I_ Classical theory-Cambridge University Press (2006).pdf", "file_path": "/home/suryaremanan/Documents/IMO/augmentoolkit/original/input/(Cambridge Studies in Advanced Mathematics 97) Hugh L. Montgomery, Robert C. Vaughan - Multiplicative number theory I_ Classical theory-Cambridge University Press (2006).pdf", "file_size": 2365282, "file_type": "application/pdf", "last_modified_date": "2025-02-03", "page_label": "227" }
228 Applications of the Prime Number Theorem 9. (Richards 1982) For every prime p ≤ y, let β( p) denote the greatest positive integer such that pβ ≤ y, and put q = ∏ p≤y p≡3 (4) p2β( p) . (a) Show that q = exp(2ψ( y;4 ,3)). (b) Show that log q ≪ y. (c) Suppose that 1 ≤ n ≤ y. Show that if n ≡ 3 (mod 4), then there is a prime p\|q such that p divides n to an odd power. (d) Let x = (q − 1)/4. Show that x is an integer, and that 4 x ≡− 1 (mod q ). (e) Show that if 1 ≤ i ≤ y/4 and p\|q , then the power of p that exactly divides x + i is the same as the power of p that exactly divides 4 i − 1. (f) Deduce that no integer in the interval ( x ,x + y/4] can be expressed as a sum of two squares. (g) Conclude that there exist arbitrarily large numbers x such that no num- ber between x and x + c log x is a sum of two squares. Here c is a suitably small positive constant. 7.4 Numbers composed of a prescribed number of primes Let σk (x ) denote the number of integers n with 1 ≤ n ≤ x and /Omega1 (n) = k. Then σ1 (x ) = π(x ) ∼ x /log x . Consider σ2 (x ). Clearly σ2 (x ) = ∑ p1 ,p2 p1 ≤ p2 p1 p2 ≤x 1 = ∑ p≤√x (π(x /p) − π( p) + O (1)) . By the Prime Number Theorem this is = ∑ p≤√x (1 + o(1)) x p(log x /p) + O (x log x ) . Thus, by partial summation and a further application of the Prime Number Theorem we ﬁnd that σ2 (x ) ∼ x log log x log x . (7.54) By inducting on k in this manner it can be shown that σk (x ) ∼ x (log log x )k−1 (k − 1)! log x (7.55)	{ "creation_date": "2025-02-03", "file_name": "(Cambridge Studies in Advanced Mathematics 97) Hugh L. Montgomery, Robert C. Vaughan - Multiplicative number theory I_ Classical theory-Cambridge University Press (2006).pdf", "file_path": "/home/suryaremanan/Documents/IMO/augmentoolkit/original/input/(Cambridge Studies in Advanced Mathematics 97) Hugh L. Montgomery, Robert C. Vaughan - Multiplicative number theory I_ Classical theory-Cambridge University Press (2006).pdf", "file_size": 2365282, "file_type": "application/pdf", "last_modified_date": "2025-02-03", "page_label": "228" }
7.4 Numbers composed of a prescribed number of primes 229 for any ﬁxed k. Since the sum over all k ≥ 1 of the right-hand side is exactly x , it is tempting to think that the above holds quite uniformly in k. However this is not the case, as we shall presently discover. T o obtain precise estimates that are uniform ink we apply analytic methods. In Section 2.4 we determined the asymptotic distribution of the additive function /Omega1 (n) − ω(n) by establishing the mean value of the multiplicative function z/Omega1 (n)−ω(n) . In the same spirit we shall derive information concerning the distribution of /Omega1 (n) from mean value estimates of z/Omega1 (n) . Since the Euler product of this latter function behaves badly when \|z\| is large, we start not with z/Omega1 (n) but with dz (n) deﬁned by the identities ζ(s)z = ∏ p ( 1 − p−s )−z = ∞∑ n=1 dz (n)n−s (σ> 1). (7.56) Since dz ( p) = z = z/Omega1 ( p) , the functions dz (n) and z/Omega1 (n) are ‘nearby’, and hence the mean value of z/Omega1 (n) can be derived from that for dz (n) by elementary reasoning. Theorem 7.17Let D z (x ) = ∑ n≤x dz (n), and let R be any positive real num- ber . If x ≥ 2, then Dz (x ) = x (log x )z−1 Ŵ(z) + O (x (log x )ℜz−2 ) uniformly for \|z\|≤ R. Proof Let a = 1 + 1/log x . Then by Corollary 5.3, Dz (x ) − 1 2πi ∫ a+iT a−iT ζ(s)z x s s ds ≪ ∑ 1 2 x <n<2x \|dz (n)\| min ( 1, x T \|x − n\| ) (7.57) + x a T ∑ n \|dz (n)\|n−a . Since \|dz (n)\| is erratic, we must exercise some care in estimating the error terms above. Let A ={ n : \|n − x \|≤ x /(log x )2 R+1 }. Without loss of generality we may suppose that R is an integer. W e note that \|dz (n)\|≤ d\|z\|(n) ≤ dR (n). By the method of the hyperbola we see by induction on R that DR (x ) = xP R (log x ) + O R ( x 1−1/R ) where PR is a polynomial of degree R − 1. Hence the contribution to the ﬁrst sum in the error term in (7.57) of the n ∈ A is ≪ ∑ n∈A \|dz (n)\|≪ x (log x )−R−2	{ "creation_date": "2025-02-03", "file_name": "(Cambridge Studies in Advanced Mathematics 97) Hugh L. Montgomery, Robert C. Vaughan - Multiplicative number theory I_ Classical theory-Cambridge University Press (2006).pdf", "file_path": "/home/suryaremanan/Documents/IMO/augmentoolkit/original/input/(Cambridge Studies in Advanced Mathematics 97) Hugh L. Montgomery, Robert C. Vaughan - Multiplicative number theory I_ Classical theory-Cambridge University Press (2006).pdf", "file_size": 2365282, "file_type": "application/pdf", "last_modified_date": "2025-02-03", "page_label": "229" }
230 Applications of the Prime Number Theorem The contribution of the n /∈ A is ≪ T −1 (log x )2 R+1 x (log x )R−1 . W e take T = exp (√ log x ) to see that this is also ≪ x (log x )−R−2 . The second sum in the error term in (7.57) is ≪ ζ(a)R ≪ (log x )R . Thus the total error term is ≪ x (log x )−R−2 . If z is a positive integer, then ζ(s)z has a pole at s = 1, and we can extract a main term by the calculus of residues, as in our proof of the Prime Number Theorem (Theorem 6.9). On the other hand, ifz is not an integer, then ζ(s)z has a branch point at s = 1, so greater care must be exercised in moving the path of integration. Put b = 1 − c/log T where c is a small positive constant, and replace the contour from a − iT to a + iT by a path consisting of C1 , C2 , C3 where C1 is a polygonal with vertices a − iT , b − iT , b − i /log x , C2 begins with a line segment from b − i /log x to 1 − i /log x , continues with the semicircle {1 + ei θ/log x : −π/2 ≤ θ ≤ π/2}, and concludes with the line segment from 1 + i /log x to b + i /log x , and ﬁnally C3 is polygonal with vertices b + i /log x , b + iT , a + iT . By Theorem 6.7, ζ(s)z ≪ (log x )R on the new path, so the integrals over C1 and C3 contribute an amount ≪ x (log x )−R−2 . On C2 we have ζ(s)z /s = (s − 1)−z (1 + O (\|s − 1\|)). Hence 1 2πi ∫ C2 ζ(s)z x s s ds = 1 2πi ∫ C2 (s − 1)−z x s ds + O (∫ C2 \|s − 1\|1−ℜz x σ \|ds \| ) . (7.58) By the change of variables s = 1 + w/log x we see that the main term above is x (log x )z−1 1 2πi ∫ H2 w−z ew d w where H2 starts at −β − i , loops around 0, and ends at −β + i where β = c(log x )/log T . Let H1 be the contour H1 ={ w = u − i : −∞ < u ≤− β}, and similarly let H3 ={ w = u + i : −∞ < u ≤− β}. If we integrate over the union of the Hi , then we obtain Hankel’s formula (see Theorem C.3) for 1 /Ŵ(z). The integral over H1 is ≪R ∫∞ β e−u/2 du ≪R e−β/2 , which is small since T = exp(√ log x ). Thus we see that the main term in (7.58) is x (log x )z−1 /Ŵ(z) + O R (x exp(−c√log x )) for some constant c. On the semi- circular part of C2 the integrand in the error term in (7.58) is ≪ x (log x )ℜz−1 ,s o the contribution is ≪ x (log x )ℜz−2 . By the change of variables s = 1 + w/log x we see that the linear portions of C2 contribute an amount ≪ x (log x )ℜz−2 ∫ ∞ 0 (u2 + 1)( R−1)/2 e−u du ≪R x (log x )ℜz−2 . Thus we have the stated estimate, and the proof is complete. □	{ "creation_date": "2025-02-03", "file_name": "(Cambridge Studies in Advanced Mathematics 97) Hugh L. Montgomery, Robert C. Vaughan - Multiplicative number theory I_ Classical theory-Cambridge University Press (2006).pdf", "file_path": "/home/suryaremanan/Documents/IMO/augmentoolkit/original/input/(Cambridge Studies in Advanced Mathematics 97) Hugh L. Montgomery, Robert C. Vaughan - Multiplicative number theory I_ Classical theory-Cambridge University Press (2006).pdf", "file_size": 2365282, "file_type": "application/pdf", "last_modified_date": "2025-02-03", "page_label": "230" }
7.4 Numbers composed of a prescribed number of primes 231 W e now establish a procedure by which we can pass from dz (n) to other nearby functions. Theorem 7.18Suppose that ∑ ∞ m=1 \|bz (m)\|(log m)2 R+1 /m is uniformly bounded for \|z\|≤ R, and for σ ≥ 1 let F (s,z) = ∞∑ m=1 bz (m)m−s . Let a z (n) be deﬁned by the relation ζ(s)z F (s,z) = ∞∑ n=1 az (n)n−s (σ> 1) and let A z (x ) = ∑ n≤x az (n). Then for x ≥ 2, Az (x ) = F (1,z) Ŵ(z) x (log x )z−1 + O ( x (log x )ℜz−2 ) . Proof Since az (n) = ∑ m\|n bz (m)dz (n/m), we see by Theorem 7.17 that Az (x ) = ∑ m≤x /2 bz (m) Dz (x /m) + ∑ x /2<m≤x bz (m) = x Ŵ(z) ∑ m≤x /2 bz (m) m (log x /m)z−1 + O ( x ∑ m≤x \|bz (m)\| m (log 2 x /m)ℜz−2 ) . (7.59) The error term here is ≪ x (log x )ℜz−2 ∑ m≤√x \|bz (m)\| m + x (log x )−R−2 ∑ m>√x \|bz (m)\| m (log m)2 R ≪ x (log x )ℜz−2 . In the main term, when m ≤ x 1/2 we write (log x /m)z−1 = (log x )z−1 + O ( (log m)(log x )ℜz−2 ) . Thus the ﬁrst sum on the right-hand side of (7.59) is = (log x )z−1 ∑ m≤x /2 bz (m) m + O ⎛ ⎝(log x )ℜz−2 ∑ m≤√ x \|bz (m)\| m log m + (log x )R−1 ∑ m>√x \|bz (m)\| m ⎞ ⎠ = (log x )z−1 F (1,z) + O ( (log x )ℜz−2 ∑ m \|bz (m)\| m (log m)2 R+1 ) ,	{ "creation_date": "2025-02-03", "file_name": "(Cambridge Studies in Advanced Mathematics 97) Hugh L. Montgomery, Robert C. Vaughan - Multiplicative number theory I_ Classical theory-Cambridge University Press (2006).pdf", "file_path": "/home/suryaremanan/Documents/IMO/augmentoolkit/original/input/(Cambridge Studies in Advanced Mathematics 97) Hugh L. Montgomery, Robert C. Vaughan - Multiplicative number theory I_ Classical theory-Cambridge University Press (2006).pdf", "file_size": 2365282, "file_type": "application/pdf", "last_modified_date": "2025-02-03", "page_label": "231" }
232 Applications of the Prime Number Theorem which gives the result. □ Suppose that R < 2, and let F (s,z) = ∏ p ( 1 − z ps )−1 ( 1 − 1 ps )z (7.60) for σ> 1, \|z\|≤ R. Then az (n) = z/Omega1 (n) in the notation of Theorem 7.18. Hence, with σk (x ) deﬁned as at the beginning of this section we ﬁnd that Az (x ) = ∑ n≤x z/Omega1 (n) = ∞∑ k=0 σk (x )zk . Here the power series on the right is actually a polynomial, since σk (x ) = 0 for sufﬁciently large k, when x is ﬁxed. Our asymptotic estimate for Az (x ) enables us to recover an estimate for the power series coefﬁcients σk (x ), since Cauchy’s formula asserts that σk (x ) = 1 2πi ∫ \|z\|=r Az (x ) zk+1 dz (7.61) for r < 2. Theorem 7.19 Suppose that R < 2, that F (s,z) is given by (7.60), and that G (z) = F (1,z)/Ŵ(z + 1). Then σk (x ) = G (k − 1 log log x )x (log log x )k−1 (k − 1)! log x ( 1 + O R ( k (log log x )2 )) (7.62) uniformly for 1 ≤ k ≤ R log log x. Since G (0) = G (1) = 1, we see that (7.55) holds when k = o(log log x ), and also when k = (1 + o(1)) log log x , but that (7.55) does not hold in general. The restriction to R < 2 is necessary because of the contribution of the prime p = 2 in the Euler product (7.60) for F (s,z). If z ≥ 2, then the behaviour is different; see Exercises 7.4.5 and 7.4.6, below . ProofOur quantitative form of the Prime Number Theorem (Theorem 6.9) gives the case k = 1, so we may assume that k > 1. W e substitute the estimate of Theorem 7.18 in (7.61) with r = (k − 1)/log log x . The error term contributes an amount ≪ x (log x )r −2r −k = x (log x )2 ek−1 (log log x )k (k − 1)k ≪ x (log log x )k (k − 1)!(log x )2 ≪ x (log log x )k−3 (k − 1)! log x .	{ "creation_date": "2025-02-03", "file_name": "(Cambridge Studies in Advanced Mathematics 97) Hugh L. Montgomery, Robert C. Vaughan - Multiplicative number theory I_ Classical theory-Cambridge University Press (2006).pdf", "file_path": "/home/suryaremanan/Documents/IMO/augmentoolkit/original/input/(Cambridge Studies in Advanced Mathematics 97) Hugh L. Montgomery, Robert C. Vaughan - Multiplicative number theory I_ Classical theory-Cambridge University Press (2006).pdf", "file_size": 2365282, "file_type": "application/pdf", "last_modified_date": "2025-02-03", "page_label": "232" }
7.4 Numbers composed of a prescribed number of primes 233 This is majorized by the error term in (7.62) since G ((k − 1)/log log x ) ≫ 1. The main term we obtain from (7.61) is xI /log x where I = 1 2πi ∫ \|z\|=r G (z)(log x )z z−k dz = G (r ) 2πi ∫ \|z\|=r (log x )z z−k dz + 1 2πi ∫ \|z\|=r (G (z) − G (r ))(log x )z z−k dz . By integration by parts we ﬁnd that r 2πi ∫ \|z\|=r (log x )z z−k dz = 1 2πi ∫ \|z\|=r (log x )z z1−k dz . W e multiply both sides by G ′(r ) and combine with the former identity to see that I = G (r ) 2πi ∫ \|z\|=r (log x )z z−k dx + 1 2πi ∫ \|z\|=r (G (z) − G (r ) − G ′(r )(z − r ))(log x )z z−k dz . (7.63) Here the ﬁrst integral is (log log x )k−1 /(k − 1)! by Cauchy’s theorem, which gives the desired main term. On the other hand, G (z) − G (r ) − G ′(r )(z − r ) = ∫ z r (z − w)G ′′(w) d w ≪\| z − r \|2 , so that if we write z = re 2πi θ, then the second integral in (7.63) is ≪ r 3−k ∫ 1/2 −1/2 (sin πθ)2 e(k−1) cos 2 πθ d θ. But \| sin x \|≤\| x \| and cos 2 πθ ≤ 1 − 8θ2 for −1/2 ≤ θ ≤ 1/2, so the above is ≪ r 3−k ek−1 ∫ ∞ 0 θ2 e−8(k−1)θ2 d θ ≪ r 3−k ek−1 (k − 1)−3/2 = (log log x )k−3 ek−1 (k − 1)k−3/2 ≪ k(log log x )k−3 /(k − 1)!. This completes the proof of the theorem. □ The decomposition in (7.63) is motivated by the observation that \|(log x )z \| is largest, for \|z\|= r , when z = r . W e take the T aylor expansion to the second term because⏐ ⏐ ⏐ ∫ (z − r )2 (log x )z z−k dz ⏐ ⏐ ⏐≍ ∫ \|z − r \|2 \|(log x )z z−k \|\| dz \|, whereas ⏐ ⏐ ⏐ ∫ (z − r )(log x )z z−k dz ⏐ ⏐ ⏐= o (∫ \|z − r \|\|(log x )z z−k \|\| dz \| ) .	{ "creation_date": "2025-02-03", "file_name": "(Cambridge Studies in Advanced Mathematics 97) Hugh L. Montgomery, Robert C. Vaughan - Multiplicative number theory I_ Classical theory-Cambridge University Press (2006).pdf", "file_path": "/home/suryaremanan/Documents/IMO/augmentoolkit/original/input/(Cambridge Studies in Advanced Mathematics 97) Hugh L. Montgomery, Robert C. Vaughan - Multiplicative number theory I_ Classical theory-Cambridge University Press (2006).pdf", "file_size": 2365282, "file_type": "application/pdf", "last_modified_date": "2025-02-03", "page_label": "233" }
234 Applications of the Prime Number Theorem By the calculus of residues we may write I = 1 (k − 1)! d k−1 dz k−1 ( G (z)(log x )z )⏐ ⏐ ⏐ z=0 = k−1∑ ν=0 G (ν) (0) ν! (log log x )k−1−ν (k − 1 − ν)! . This gives a more accurate, but more complicated, main term. In Section 2.3 we saw that /Omega1 (n) rarely differs very much from log log n. In particular, from Theorem 2.12 we see that if r < 1, then the number of n ≤ x for which /Omega1 (n) <r log log n is ≪r x /log log x .W en o wg i v ea much sharper upper bound for the number of occurrences of such large deviations. Theorem 7.20Let A (x ,r ) denote the number of n ≤ x such that /Omega1 (n) ≤ r log log x , and let B (x ,r ) denote the number of n ≤ x for which /Omega1 (n) ≥ r log log x. I f 0 <r ≤ 1 and x ≥ 2, then A(x ,r ) ≪ x (log x )r −1−r log r . If 1 ≤ r ≤ R < 2 and x ≥ 2, then B (x ,r ) ≪R x (log x )r −1−r log r . Proof W e argue directly from Theorem 7.18, using a modiﬁed form of Rankin’s method. If 0 ≤ r ≤ 1 and /Omega1 (n) ≤ r log log x , then r r log log x ≤ r /Omega1 (n) . Hence A(x ,r ) ≤ (log x )−r log r ∑ n≤x r /Omega1 (n) . By Theorem 7.18 this is ∼ F (1,r ) Ŵ(r ) x (log x )r −1−r log r where F (s,z) is taken as in (7.60). This gives the result since F (1,r ) ≪ 1 and Ŵ(r ) ≫ 1 uniformly for 0 <r ≤ 1. Now suppose that 1 ≤ r ≤ R < 2 and that /Omega1 (n) ≥ r log log x . Then r /Omega1 (n) ≥ r r log log x , and hence B (x ,r ) ≤ (log x )−r log r ∑ n≤x r /Omega1 (n) . Thus we have only to proceed as before to obtain the result. □	{ "creation_date": "2025-02-03", "file_name": "(Cambridge Studies in Advanced Mathematics 97) Hugh L. Montgomery, Robert C. Vaughan - Multiplicative number theory I_ Classical theory-Cambridge University Press (2006).pdf", "file_path": "/home/suryaremanan/Documents/IMO/augmentoolkit/original/input/(Cambridge Studies in Advanced Mathematics 97) Hugh L. Montgomery, Robert C. Vaughan - Multiplicative number theory I_ Classical theory-Cambridge University Press (2006).pdf", "file_size": 2365282, "file_type": "application/pdf", "last_modified_date": "2025-02-03", "page_label": "234" }
7.4 Numbers composed of a prescribed number of primes 235 In discussing Theorem 2.12 we proposed a probabilistic model, which in conjunction with the Central Limit Theorem would predict that the quantity αn = /Omega1 (n) − log log n√ log log n (7.64) is asymptotically normally distributed. W e now conﬁrm this. Theorem 7.21Let αn be given by (7.64) and suppose that Y > 0. Then the number of n, 3 ≤ n ≤ x , such that αn ≤ yi s /Phi1 ( y)x + OY ( x √ log log x ) uniformly for −Y ≤ y ≤ Y where /Phi1 ( y) = 1√ 2π ∫ y −∞ e−t 2 /2 dt . Proof Let βn = /Omega1 (n) − log log x√ log log x . Since /Phi1 ′( y) ≪ 1 and αn − βn ≪ 1/√ log log x when x 1/2 ≤ n ≤ x and /Omega1 (n) ≤ 2 log log x , it sufﬁces to consider βn in place of αn . W e may of course also suppose that x is large. Let k be a natural number and let u be deﬁned by writing k = u + log log x . If \|u\|≤ 1 2 log log x , then by Stirling’s formula (see (B.26) or the more general Theorem C.1) we see that (log log x )k−1 (k − 1)! = eu log x√2π log log x ( 1 + u log log x )1 2 −log log x −u ( 1 + O ( 1 log log x )) . The estimate log(1 + δ) = δ − δ2/2 + O (\|δ\|3 ) holds uniformly for \|δ\|≤ 1/2. By taking δ = u/log log x we ﬁnd that ( 1 + u log log x )1 2 −log log x −u = exp ( −u + u − u2 2 log log x − u2 4(log log x )2 + O ( \|u\|3 (log log x )2 )) .	{ "creation_date": "2025-02-03", "file_name": "(Cambridge Studies in Advanced Mathematics 97) Hugh L. Montgomery, Robert C. Vaughan - Multiplicative number theory I_ Classical theory-Cambridge University Press (2006).pdf", "file_path": "/home/suryaremanan/Documents/IMO/augmentoolkit/original/input/(Cambridge Studies in Advanced Mathematics 97) Hugh L. Montgomery, Robert C. Vaughan - Multiplicative number theory I_ Classical theory-Cambridge University Press (2006).pdf", "file_size": 2365282, "file_type": "application/pdf", "last_modified_date": "2025-02-03", "page_label": "235" }
236 Applications of the Prime Number Theorem Suppose now that \|u\|≤ (log log x )2/3 . By considering separately \|u\|≤ (log log x )1/2 and (log log x )1/2 < \|u\|≤ (log log x )2/3 we see that u log log x ≪ 1√ log log x + \|u\|3 (log log x )2 . Similarly , by considering \|u\|≤ 1 and \|u\| > 1 we see that u2 (log log x )2 ≪ 1√ log log x + \|u\|3 (log log x )2 . On combining these estimates we deduce that (log log x )k−1 (k − 1)! = log x√2π log log x exp ( −u2 2 log log x ) × ( 1 + O ( 1√ log log x ) + O ( \|u\|3 (log log x )2 )) uniformly for \|u\|≤ (log log x )2/3 . In Theorem 7.19 we have G (1) = 1 and G (k − 1 log log x ) = G (1) + O (1 +\| u\| log log x ) . Hence by Theorem 7.19, σk (x ) = x exp ( −(k−log log x )2 2 log log x ) √2π log log x × ( 1 + O ( 1√ log log x ) + O (\|k − log log x \|3 (log log x )2 )) . By Theorem 7.20 we know that the contribution of k ≤ log log x − (log log x )2/3 is negligible. W e sum over the range log log x − (log log x )2/3 ≤ k ≤ log log x + y(log log x )1/2 . This gives rise to three sums, one for the main term and two for error terms. Each of these sums can be considered to be a Riemann sum for an associated integral, and the stated result follows.□ 7.4.1 Exercises 1. Let p1 , p2 ,..., pK be distinct primes. Show that the number of n ≤ x composed entirely of the pk is (log x )K K ! ∏ K k=1 log pk + O ( (log x )K −1 ) .	{ "creation_date": "2025-02-03", "file_name": "(Cambridge Studies in Advanced Mathematics 97) Hugh L. Montgomery, Robert C. Vaughan - Multiplicative number theory I_ Classical theory-Cambridge University Press (2006).pdf", "file_path": "/home/suryaremanan/Documents/IMO/augmentoolkit/original/input/(Cambridge Studies in Advanced Mathematics 97) Hugh L. Montgomery, Robert C. Vaughan - Multiplicative number theory I_ Classical theory-Cambridge University Press (2006).pdf", "file_size": 2365282, "file_type": "application/pdf", "last_modified_date": "2025-02-03", "page_label": "236" }
7.4 Numbers composed of a prescribed number of primes 237 2. (a) Let dz (n) be deﬁned as in (7.56), and suppose that \|z\|≤ R. Show that \|dz (n)\|≤ d\|z\|(n) ≤ dR (n). (b) Let F (s,z) be deﬁned as in (7.60). Show that if 0 <r < 1 and σ> 1/2, then 0 < F (σ,r ) < 1. (c) Let F (s,z) be deﬁned as in (7.60). Show that if 1 <r < 2, then the Dirichlet series coefﬁcients of F (s,r ) are all non-negative. 3. (a) Show that if F (s,z) = ∏ p ( 1 + z ps − 1 )( 1 − 1 ps )z , then F (s,z) converges for σ> 1/2, uniformly for \|z\|≤ R. (b) Show that if F (s,z) is taken as above, and if az (n) is deﬁned as in Theorem 7.18, then az (n) = zω(n) . (c) Let ρk (x ) denote the number of n ≤ x for which ω(n) = k. Show that if x ≥ 2, then ρk (x ) = G (k − 1 log log x )x (log log x )k−1 (k − 1)! log x ( 1 + O R ( k (log log x )2 )) uniformly for 1 ≤ k ≤ R log log x where G (z) = F (1,z)/Ŵ(z + 1). (d) Show that G (0) = G (1) = 1. (e) Let A(x ,r ) denote the number of n ≤ x for which ω(n) ≤ r log log x . Show that A(x ,r ) ≪ x (log x )r −1−r log r uniformly for 0 <r ≤ 1. (f) Let B (x ,r ) denote the number of n ≤ x for which ω(n) ≥ r log log x . Show that B (x ,r ) ≪ x (log x )r −1−r log r uniformly for 1 ≤ r ≤ R. 4. (a) Show that if F (s,z) = ∏ p ( 1 + z ps )( 1 − 1 ps )z , then F (s,z) converges for σ> 1/2, uniformly for \|z\|≤ R. (b) Show that if F (s,z) is taken as above, and if az (n) is deﬁned as in Theorem 7.18, then az (n) = µ(n)2 zω(n) . (c) Let πk (x ) denote the number of square-free n ≤ x for which ω(n) = k. Show that if x ≥ 2, then πk (x ) = G (k − 1 log log x )x (log log x )k−1 (k − 1)! log x ( 1 + O R ( k (log log x )2 )) uniformly for 1 ≤ k ≤ R log log x where G (z) = F (1,z)/Ŵ(z + 1).	{ "creation_date": "2025-02-03", "file_name": "(Cambridge Studies in Advanced Mathematics 97) Hugh L. Montgomery, Robert C. Vaughan - Multiplicative number theory I_ Classical theory-Cambridge University Press (2006).pdf", "file_path": "/home/suryaremanan/Documents/IMO/augmentoolkit/original/input/(Cambridge Studies in Advanced Mathematics 97) Hugh L. Montgomery, Robert C. Vaughan - Multiplicative number theory I_ Classical theory-Cambridge University Press (2006).pdf", "file_size": 2365282, "file_type": "application/pdf", "last_modified_date": "2025-02-03", "page_label": "237" }
238 Applications of the Prime Number Theorem (d) Show that G (0) = G (1) = 1. 5. (a) Show that if x ≥ 2, then ∑ n≤x 2/Omega1 (n) = cx (log x )2 + O (x log x ) where c is a positive constant. (b) Show that if x ≥ 2, then ∑ n≤x 2ω(n) = cx log x + O (x ) where c is a positive constant. 6. Show that if (2 + ε) log log x ≤ k ≤ R log log x , then σk (x ) ∼ c2−k x log x . 7. Show that if δ ≤ r ≤ 1 − δ (or 1 + δ ≤ r ≤ 2 − δ), then A(x ,r ) (or B (x ,r ), respectively) is ≍ x (log x )r −1−r log r /√ log log r . 8. Show that if x is large, then there is a k such that σk (x ) ≥ x 3√ log log x . 9. Show that the mean value ∑ n≤x d (n) ∼ x log x is due to the numbers n ≤ x for which \|ω(n) − 2 log log x \|≪ √ log log x . 10. Suppose that 1 /2 ≤ r ≤ R. Show that the number of square-free n ≤ x that can be written as a sum of two squares and for which ω(n) ≥ r log log x is ≪R x (log x )r −1−r log 2 r . 11. (Addison 1957) Let Mq ,k (x ) denote the number of n ≤ x such that /Omega1 (n) ≡ k (mod q ). (a) Show that if q is ﬁxed, then Mq ,k (x ) ∼ x /q as x →∞ . (b) Show that if q is ﬁxed, q > 2, then Mq ,k (x ) − x q = /Omega1 ± ( x (log x )κ ) where κ = 1 − cos 2 π/q . 12. Show that ∑ 1<n≤x 1 ω(n) ∼ x log log x as x →∞ . 13. Show that if x ≥ 2, then ∑ 1<n≤x /Omega1 (n) ω(n) = x + O ( x log log x ) .	{ "creation_date": "2025-02-03", "file_name": "(Cambridge Studies in Advanced Mathematics 97) Hugh L. Montgomery, Robert C. Vaughan - Multiplicative number theory I_ Classical theory-Cambridge University Press (2006).pdf", "file_path": "/home/suryaremanan/Documents/IMO/augmentoolkit/original/input/(Cambridge Studies in Advanced Mathematics 97) Hugh L. Montgomery, Robert C. Vaughan - Multiplicative number theory I_ Classical theory-Cambridge University Press (2006).pdf", "file_size": 2365282, "file_type": "application/pdf", "last_modified_date": "2025-02-03", "page_label": "238" }
7.5 Notes 239 14. Suppose that 0 ≤ α ≤ 1. Show that ∑ n≤x card{m : m\|n,m ≤ nα} d (n) = 2 πx arcsin √α + O ( x√ log x ) . 15. Show that if x ≥ 16, then ∑ n≤x (n,/Omega1 (n))=1 1 = 6 π2 x + O ( x log log log x ) . 7.5 Notes Section 7.1. Theorem 7.2 was ﬁrst proved by Dickman (1930), and was redis- covered by Chowla & V ijayaraghavan (1947), Ramaswami (1949), and Buch- stab (1949). de Bruijn (1951a) gave a more precise estimate forψ(x ,y), over a longer range of y. There is a considerable range of applications of ψ(x ,y), such as those to the distribution of kth power residues, W aring’s problem, and the complexity of arithmetical algorithms in computer science. As a reﬂection of this there have been two signiﬁcant survey articles, by Norton (1971) and by Hildebrand & T enenbaum (1993). Our treatment of ψ(x ,y) is fairly elementary , but it would be natural to take a more analytic approach, and use Perron’s formula to write ψ(x ,y) = 1 2πi ∫ c+i ∞ c−i ∞ ∏ p≤y (1 − p−s )−1 x s s ds = 1 2πi ∫ c+i ∞ c−i ∞ ζ(s) ∏ p>y (1 − p−s ) x s s ds . For s not too large, an approximation to the product over p > y is provided by the Prime Number Theorem, and this suggests the main term /Lambda1 (x ,y) = 1 2πi ∫ c+i ∞ c−i ∞ ζ(s)e x p ( − ∫ ∞ y v−s (log v)−1 d v )x s s ds . It can be shown that this is indeed a good approximation to ψ(x ,y)o v e rav e r y long range, but the technical details are rather heavy . By Theorem 7.10 it is not hard to show that /Lambda1 (x ,y) = x ∫ ∞ 0− ρ(u − v)d ([ yv] y−v) where we use (7.30) to extend the deﬁnition of ρ(u)t o u ≤ 0. It follows that /Lambda1 (x ,y) ∼ ρ(u)x	{ "creation_date": "2025-02-03", "file_name": "(Cambridge Studies in Advanced Mathematics 97) Hugh L. Montgomery, Robert C. Vaughan - Multiplicative number theory I_ Classical theory-Cambridge University Press (2006).pdf", "file_path": "/home/suryaremanan/Documents/IMO/augmentoolkit/original/input/(Cambridge Studies in Advanced Mathematics 97) Hugh L. Montgomery, Robert C. Vaughan - Multiplicative number theory I_ Classical theory-Cambridge University Press (2006).pdf", "file_size": 2365282, "file_type": "application/pdf", "last_modified_date": "2025-02-03", "page_label": "239" }
240 Applications of the Prime Number Theorem for a large range of u. For the further development of the theory , especially on the analytic side, see Hildebrand & T enenbaum (1993). Section 7.2. Theorem 7.11 is due to Buchstab (1937). The ﬁner details of the behaviour of /Phi1 (x ,y) when u is large are intimately connected with sieve theory , especially that of the linear sieve, i.e., the sieve in which on average one residue class (modp) is removed. The standard references are Greaves (2001), Halberstam & Richert (1974), Selberg (1991). Section 7.3. Theorem 7.14 was ﬁrst proved by W estzynthius (1931). Erd ˝ os (1935a) showed that lim sup n→∞ pn+1 − pn (log pn )(log log pn )/(log log log pn )2 > 0, and then Rankin (1938) obtained Theorem 7.15 with c = 1/3. The value of c has been successively improved by Sch ¨ onhage (1963), Rankin (1963), Maier & Pomerance (1990), culminating in the valuec = 2eC0 of Pintz (1997). Erd ˝ os offered a $10,000 prize for the ﬁrst proof that the limsup in Theorem 7.15 is +∞. Early studies of g( P (z)) were conducted by Backlund (1929), Brauer & Zeitz (1930), Ricci (1934), and Chang (1938). The size of g( P (z)) is not known; possibly it is ≍ z log z. However, it is conceivable that inﬁnitely often pn+1 − pn is as large as (log pn )θ where θ> 1. In particular, Cram´ er (1936) conjectured that lim sup n→∞ pn+1 − pn (log pn )2 = 1. Theorem 7.16 is due to Hensley & Richards (1973). Section 7.4. The analysis ofσk (x ) is based on Selberg’s exposition (1954) of Sathe (1953a,b, 1954a,b). Sathe (1954b) also shows that the bound R log log x cannot be replaced by 2 log log x + 1. Arguments giving rise to versions of Theorem 7.20 occur in Erd ˝ os (1935b). A qualitative version of Theorem 7.21 is a special case of Erd ˝ os & Kac (1940). Quantitative versions with various weaker error terms were obtained by LeV eque (1949) and Kubilius (1956). Theorem 7.21 had been conjectured by LeV eque and was established by R´ enyi & Tur´ an (1958). They also showed that the error term is both uniform inx and best-possible. 7.6 References Addison, A. W . (1957). A note on the compositeness of numbers, Proc. Amer . Math. Soc. 8, 151–154.	{ "creation_date": "2025-02-03", "file_name": "(Cambridge Studies in Advanced Mathematics 97) Hugh L. Montgomery, Robert C. Vaughan - Multiplicative number theory I_ Classical theory-Cambridge University Press (2006).pdf", "file_path": "/home/suryaremanan/Documents/IMO/augmentoolkit/original/input/(Cambridge Studies in Advanced Mathematics 97) Hugh L. Montgomery, Robert C. Vaughan - Multiplicative number theory I_ Classical theory-Cambridge University Press (2006).pdf", "file_size": 2365282, "file_type": "application/pdf", "last_modified_date": "2025-02-03", "page_label": "240" }
7.6 References 241 Alladi, K. & Erd ˝ os, P . (1977). On an additive arithmetic function, P aciﬁc J. Math . 71, 275–294. Backlund, R. J. (1929). ¨Uber die Differenzen zwischen den Zahlen, die zu den n ersten Primzahlen teilerfremd sind, Annales Acad. sci. F ennicae 32 (Lindel ¨ of-Festschrift), Nr. 2, 9 pp. Brauer, A. & Zeitz, H. (1930). ¨Uber eine zahlentheoretische Behauptung von Legendre, Sitzungsb. Math. Ges. Berlin 29, 116–125. de Bruijn, N. G. (1949). The asymptotically periodic behavior of the solutions of some linear functional equations, Amer . J. Math. 71, 313–330. (1950a). On the number of uncancelled elements in the sieve of Eratosthenes, Nederl. Akad. W etensch. Proc. 52, 803–812. ( = Indag. Math. 12, 247–256) (1950b). On some linear functional equations, Publ. Math. 1, 129–134. (1951a). The asymptotic behaviour of a function occurring in the theory of primes, J. Indian Math. Soc. 15 (A), 25–32. (1951b). On the number of positive integers ≤ x and free of prime factors > y, Proc. Nederl. Akad. W etensch. 54, 50–60. (1966). On the number of positive integers ≤ x and free of prime factors > y, II, Proc. Koninkl. Nederl. Akad. W etensch .A 69, 239–247. ( = Indag. Math. 28) Buchstab, A. A. (1937). Asymptotic estimates of a general number-theoretic function, Mat. Sb. (2) 44, 1239–1246. (1949). On those numbers in an arithmetic progression all prime factors of which are small in magnitude, Dokl. Akad. Nauk SSSR (N. S.) 67, 5–8. Chang, T .-H. (1938). ¨Uber aufeinanderfolgende Zahlen, von denen jede mindestens einer von n linearen Kongruenzen gen ¨ ugt, deren Moduln die ersten n Primzahlen sind, Schr . Math. Sem. Inst. Angew . Math. Univ . Berlin 4, 35–55. Chowla, S. D. & V ijayaraghavan, T . (1947). On the largest prime divisors of numbers, J. Indian Math. Soc. (2) 12, 31–37. Cram´ er, H. (1936). On the order of magnitude of the difference between consecutive prime numbers, Acta Arith . 2, 23–46. DeKoninck, J.-M. (1972). On a class of arithmetical functions, Duke Math. J. 39, 807– 818. Dickman, K. (1930). On the frequency of numbers containing prime factors of a certain relative magnitude, Ark. Mat. Astr . fys. 22, 1–14. Duncan, R. L. (1970). On the factorization of integers, Proc. Amer . Math. Soc. 25, 191–192. Erd ˝ os, P . (1935a). On the difference of consecutive primes, Quart. J. Math., Oxford ser . 6, 124–128. (1935b). On the normal number of prime factors of p − 1 and some related problems concerning Euler’s φ- function. Quart. J. Math., Oxford ser . 6, 205–213. (1946). Some remarks about additive and multiplicative functions, Bull. Amer . Math. Soc. 52, 527–537. (1951). Some problems and results in elementary number theory , Publ. Math. Debre- cen 2, 103–109. (1962). On the integers relatively prime to n and on a number-theoretic function considered by Jacobsthal, Math. Scand. 10, 163–170. (1963). Problem and Solution Nr. 136, Wiskundige opgaven met de Oplossingen 21.	{ "creation_date": "2025-02-03", "file_name": "(Cambridge Studies in Advanced Mathematics 97) Hugh L. Montgomery, Robert C. Vaughan - Multiplicative number theory I_ Classical theory-Cambridge University Press (2006).pdf", "file_path": "/home/suryaremanan/Documents/IMO/augmentoolkit/original/input/(Cambridge Studies in Advanced Mathematics 97) Hugh L. Montgomery, Robert C. Vaughan - Multiplicative number theory I_ Classical theory-Cambridge University Press (2006).pdf", "file_size": 2365282, "file_type": "application/pdf", "last_modified_date": "2025-02-03", "page_label": "241" }
242 Applications of the Prime Number Theorem Erd ˝ os, P . & Kac, M. (1940). The Gaussian law of errors in the theory of additive number theoretic functions, Amer . J. Math. 62, 738–742. Erd ˝ os, P . & Nicolas, J.-L. (1981). Sur la fonction: nombre de facteurs premiers de n, Enseignoment Math. (2) 27, 3–27. Friedlander, J. B. (1972). Maximal sets of integers with small common divisors, Math. Ann. 195, 107–113. Greaves, G. (2001). Sieves in Number Theory , Ergeb. Math. (3) 43. Berlin: Springer- V erlag. Halberstam, H. (1970). On integers all of whose prime factors are small, Proc. London Math. Soc. (3) 21, 102–107. Halberstam, H. & Richert, H.-E. (1974). Sieve Methods , London Mathematical Society Monographs No. 4. London: Academic Press, 1974. Hardy , G. H. & Littlewood, J. E. (1923). Some problems of “Partitio Numerorum”: III On the expression of a number as a sum of primes, Acta Math. 44, 1–70. Hausman, M. & Shapiro, H. N. (1973). On the mean square distribution of primitive roots of unity, Comm. Pure Appl. Math. 26, 539–547. Hensley , D. & Richards, I. (1973). T wo conjectures concerning primes, Analytic Number Theory , Proc. Sympos. Pure Math. 24. Providence: Amer. Math. Soc., 123–128. (1973/4). Primes in intervals, Acta Arith. 25, 375–391. Hildebrand, A. (1984). Integers free of large prime factors and the Riemann Hypothesis, Mathematika 31, 258–271. (1985). Integers free of large prime divisors in short intervals, Oxford Quart. J. 36, 57–69. (1986a). On the number of positive integers ≤ x and free of prime factors > y, J. Number Theory 22, 289–307. (1986b). On the local behavior of ψ(x ,y), Trans. Amer . Math. Soc. 297, 729–751. (1987). On the number of prime factors of integers without large prime divisors, J. Number Theory 25, 81–106. Hildebrand, A. & T enenbaum, G. (1986). On integers free of large prime factors, Trans. Amer . Math. Soc. 296, 265–290. (1993). Integers without large prime factors, J. Th ´eor . Nombres Bordeaux. 5, 411–484. Kubilius, I. P . (1956). Probabilistic methods in the theory of numbers, Uspehi Mat. Nauk (N.S.) 11 68, 31–66. Legendre, A. M. (1798). Th´eorie des Nombres , First edition, V ol. 2, pp. 71–79. LeV eque, W . J. (1949). On the size of certain number-theoretic functions, Trans. Amer . Math. Soc. 66, 440–463. Maier, H. & Pomerance, C. (1990). Unusually large gaps between consecutive primes, Trans. Amer . Math. Soc. 322, 201–237. Montgomery , H. L. (1987). Fluctuations in the mean of Euler’s phi function, Proc. Indian Acad. Sci. (Math. Sci.) 97, 239–245. Montgomery , H. L. & V aughan, R. C. (1986). On the distribution of reduced residues, Ann. of Math. (2) 123 (1986), 311–333. Norton, K. K. (1971). Numbers with Small F actors and the Least k ’th P ower Non- Residues, Memoir 106, Providence: Amer. Math. Soc. Pillai, S. S. & Chowla, S. D. (1930). On the error terms in some asymptotic formulæ in the theory of numbers, I, J. London Math Soc. 5, 95–101.	{ "creation_date": "2025-02-03", "file_name": "(Cambridge Studies in Advanced Mathematics 97) Hugh L. Montgomery, Robert C. Vaughan - Multiplicative number theory I_ Classical theory-Cambridge University Press (2006).pdf", "file_path": "/home/suryaremanan/Documents/IMO/augmentoolkit/original/input/(Cambridge Studies in Advanced Mathematics 97) Hugh L. Montgomery, Robert C. Vaughan - Multiplicative number theory I_ Classical theory-Cambridge University Press (2006).pdf", "file_size": 2365282, "file_type": "application/pdf", "last_modified_date": "2025-02-03", "page_label": "242" }
7.6 References 243 Pintz, J. (1997). V ery large gaps between consecutive primes, J. Number Theory 63, 286–301. Ramaswami, V . (1949). The number of positive integers ≤ x and free of prime divisors > y, and a problem of S. S. Pillai, Duke Math. J. 16, 99–109. Rankin, R. A. (1938). The difference between consecutive primes, J. London Math. Soc. 13, 242–247. (1963). The difference between consecutive primes, V, Proc. Edinburgh Math. Soc. (2)13, 331–332. R´enyi, A. & Tur´ an, P . (1958), On a theorem of Erd ˝ os–Kac, Acta Arith . 4, 71–84. Ricci, G. (1934). Ricerche aritmetiche sui polinomi , II, Rend. P alermo 58, 190–208. Richards, I. (1982). On the gaps between numbers which are sums of two squares, Adv . in Math. 46, 1–2. Sathe, L. G. (1953a,b,1954a,b). On a problem of Hardy on the distribution of integers with a given number of prime factors I, II, III, IV, J. Indian Math. Soc. (N.S.) 17, 63–82 & 83–141, 18, 27–42 & 43–81. Schinzel, A. (1961). Remarks on the paper “Sur certaines hypoth` eses concernant les nombres premiers”, Acta Arith. 7, 1–8. Sch ¨ onhage, A. (1963). Eine Bemerkung zur Konstruktion grosser Primzahll ¨ ucken, Arch. Math. 14, 29–30. Selberg, A. (1954). Note on a paper of L. G. Sathe, J. Indian Math. Soc. 18, 83–87. (1991). Collected papers , V ol. II. Berlin: Springer-V erlag. W estzynthius, E. (1931). ¨Uber die V erteilung der Zahlen, die zu den n ersten Primzahlen teilerfremd sind, Comment. Phys.–Math. Soc. Sci. F ennica 5, Nr. 25, 37 pp.	{ "creation_date": "2025-02-03", "file_name": "(Cambridge Studies in Advanced Mathematics 97) Hugh L. Montgomery, Robert C. Vaughan - Multiplicative number theory I_ Classical theory-Cambridge University Press (2006).pdf", "file_path": "/home/suryaremanan/Documents/IMO/augmentoolkit/original/input/(Cambridge Studies in Advanced Mathematics 97) Hugh L. Montgomery, Robert C. Vaughan - Multiplicative number theory I_ Classical theory-Cambridge University Press (2006).pdf", "file_size": 2365282, "file_type": "application/pdf", "last_modified_date": "2025-02-03", "page_label": "243" }
8 Further discussion of the Prime Number Theorem 8.1 Relations equivalent to the Prime Number Theorem The Prime Number Theorem asserts that π(x ) ∼ x log x (8.1) as x →∞ . In this section we consider a number of asymptotic relations that are equivalent to the Prime Number Theorem in the sense that they can be derived from, and also imply the Prime Number Theorem, by means of simple elementary arguments. These relations can also be proved by using the same analytic machinery that we used to prove the Prime Number Theorem, but the elementary techniques that we use to derive one relationship from another have permanent utility . In Corollary 2.5 we saw that π(x ) = ψ(x )/log x + O (x /(log x )2 ) and that ψ(x ) = ϑ(x ) + O ( x 1/2 ) . Hence (8.1) is equivalent to ψ(x ) = x + o(x ), (8.2) and also to ϑ(x ) = x + o(x ). (8.3) These equivalences are fairly trivial, since the arithmetic functions involved are nearly the same. At a somewhat deeper level, we considerM (x ) = ∑ n≤x µ(n), and show that the estimate M (x ) = o(x ) (8.4) is equivalent to the Prime Number Theorem. As was remarked in Chapter 6, the relation (8.4) can be proved analytically , by applying the truncated Perron formula to the Dirichlet series 1/ζ(s) and using the zero-free region of the zeta function, as in the proof of the Prime Number Theorem. T o derive (8.4) from 244	{ "creation_date": "2025-02-03", "file_name": "(Cambridge Studies in Advanced Mathematics 97) Hugh L. Montgomery, Robert C. Vaughan - Multiplicative number theory I_ Classical theory-Cambridge University Press (2006).pdf", "file_path": "/home/suryaremanan/Documents/IMO/augmentoolkit/original/input/(Cambridge Studies in Advanced Mathematics 97) Hugh L. Montgomery, Robert C. Vaughan - Multiplicative number theory I_ Classical theory-Cambridge University Press (2006).pdf", "file_size": 2365282, "file_type": "application/pdf", "last_modified_date": "2025-02-03", "page_label": "244" }
8.1 Relations equivalent to the Prime Number Theorem 245 (8.2) it would be natural to express µ(n) as the Dirichlet convolution of /Lambda1 (n) with some other function. As an aid to discovering such a function we would write 1 ζ(s) = ζ′(s) ζ(s) · 1 ζ′(s) . Unfortunately , 1 /ζ′(s) =− 1/∑ (log n)n−s cannot be expanded as a Dirichlet series (because log 1 = 0), so we reach an impasse. T o circumvent this difﬁculty we introduce a valuable trick. Instead of treating M (x ) directly , we ﬁrst consider N (x ): = ∑ n≤x µ(n) log n. Since M (x ) log x − N (x ) = ∑ n≤x µ(n) log( x /n) ≪ ∑ n≤x log(x /n) ≪ x , it is clear that (8.4) is equivalent to the estimate N (x ) = o(x log x ). (8.5) T o derive (8.5) from (8.2) we observe that the Dirichlet series generating func- tion ofµ(n) log n is −(1/ζ(s))′ = ζ′(s)/ζ(s)2 . Alternatively , in elementary lan- guage, we recall (1.22), which asserts that ∑ d \|n /Lambda1 (d ) = log n ( − ζ′ ζ (s) · ζ(s) =− ζ′(s) ) . By the M ¨ obius inversion formula, this gives /Lambda1 (n) = ∑ d \|n µ(d ) log n/d ( − ζ′ ζ (s) =− ζ′(s) · 1/ζ(s) ) , (8.6) as was already noted in the proof of Theorem 2.4. But 0 = (log n) ∑ d \|n µ(d ) ( 0 = d ds (ζ(s) · 1/ζ(s)) ) for all n, and so /Lambda1 (n) =− ∑ d \|n µ(d ) log d ( − ζ′ ζ (s) =− ζ(s) · (ζ′(s)/ζ(s)2 ) ) . By M ¨ obius inversion a second time, we deduce that µ(n) log n =− ∑ d \|n µ(d )/Lambda1 (n/d ) ( ζ′(s)/ζ(s)2 = (1/ζ(s)) · ζ′ ζ (s) ) . Since /Lambda1 (n/d ) is 1 on average, we adjust by this amount: ∑ d \|n µ(d )(1 − /Lambda1 (n/d )) = {µ(n) log n (n > 1), 1( n = 1).	{ "creation_date": "2025-02-03", "file_name": "(Cambridge Studies in Advanced Mathematics 97) Hugh L. Montgomery, Robert C. Vaughan - Multiplicative number theory I_ Classical theory-Cambridge University Press (2006).pdf", "file_path": "/home/suryaremanan/Documents/IMO/augmentoolkit/original/input/(Cambridge Studies in Advanced Mathematics 97) Hugh L. Montgomery, Robert C. Vaughan - Multiplicative number theory I_ Classical theory-Cambridge University Press (2006).pdf", "file_size": 2365282, "file_type": "application/pdf", "last_modified_date": "2025-02-03", "page_label": "245" }
246 Further discussion of the Prime Number Theorem W e sum this over n ≤ x (which is to say we apply (2.7)) to see that ∑ d ≤x µ(d )([x /d ] − ψ(x /d )) = N (x ) + 1. From (8.2) we know that for any ε> 0 there is a large number C = C (ε) such that \|ψ( y) − [ y]\| <ε y provided that y ≥ C . That is, \|ψ(x /d ) − [x /d ]\|≤ εx /d for d ≤ x /C . Thus⏐ ⏐ ⏐ ⏐ ⏐ ∑ d ≤x /C µ(d ) (ψ(x /d ) − [x /d ]) ⏐ ⏐ ⏐ ⏐ ⏐≤ ∑ d ≤x /C εx d ≪ εx log x . The remaining range we treat trivially: ∑ x /C <d ≤x µ(d )(ψ(x /d ) − [x /d ]) ≪ ∑ x /C <d ≤x x d ≪ x log 2 C. Since εcan be taken arbitrarily small, we see that (8.5), and hence (8.4), follows from (8.2). It is worth pausing here to note that the choice of the main term above is extremely delicate. If we had subtracted x /d instead of [ x /d ], then we would have had to consider the question of the size of the sum ∑ d ≤x µ(d )/d , which will be considered later. Since ∑ d ≤x µ(d )[x /d ] = 1 for all x ≥ 1, we avoid the problem by this judicious choice of the main term. T o complete our proof that (8.4) is equivalent to (8.2) we now assume (8.4), and derive (8.2). By summing (8.6) over n, which is to say by applying (2.7), we see that ψ(x ) = ∑ d ≤x µ(d )T (x /d ) where T (x ) = ∑ m≤x log m as in Section 2.2. W e recall that T (x ) = x log x − x + O (log x ) by the integral test. The main term here is approximately the same as applies to the summatory function of the divisor function, since Theorem 2.2 asserts thatD(x ) = ∑ m≤x d (m) = x log x + (2C0 − 1)x + O ( x 1/2 ) . Indeed, the arithmetic function d (m) − 2C0 , when summed over m, produces exactly the same main terms as log m. That is, if f (m) = log m − d (m) + 2C0 and F (x ) = ∑ m≤x f (m) then F (x ) ≪ x 1/2 . On the other hand, ∑ r \|n µ(r )d (n/r ) = 1 for all n and ∑ d \|n µ(d ) = 0 for all n > 1, so that ∑ d \|n µ(d ) f (n/d ) = {/Lambda1 (n) − 1( n > 1), 2C0 − 1( n = 1). On summing this over n ≤ x we ﬁnd that ∑ d ≤x µ(d ) F (x /d ) = ψ(x ) − [x ] + 2C0 . (8.7)	{ "creation_date": "2025-02-03", "file_name": "(Cambridge Studies in Advanced Mathematics 97) Hugh L. Montgomery, Robert C. Vaughan - Multiplicative number theory I_ Classical theory-Cambridge University Press (2006).pdf", "file_path": "/home/suryaremanan/Documents/IMO/augmentoolkit/original/input/(Cambridge Studies in Advanced Mathematics 97) Hugh L. Montgomery, Robert C. Vaughan - Multiplicative number theory I_ Classical theory-Cambridge University Press (2006).pdf", "file_size": 2365282, "file_type": "application/pdf", "last_modified_date": "2025-02-03", "page_label": "246" }
8.1 Relations equivalent to the Prime Number Theorem 247 W e now use (8.4) to show that the left-hand side above is o(x ), which thus gives (8.2). The reasoning employed at this point is useful for other purposes, so we axiomatize the argument, as follows. Theorem 8.1(Axer’s theorem) Suppose that a d is a sequence such that (i) ∑ d ≤x ad = o(x ) and that (ii) ∑ d ≤x \|ad \|≪ x . Suppose also that F (x ) is a function deﬁned on [1,∞) such that (iii) F (x ) has bounded variation in the interval [1,C ] for any ﬁnite C ≥ 1, and that (iv) F (x ) ≪ x /(log x )c for some constant c > 1. Then ∑ d ≤x ad F (x /d ) = o(x ). By taking ad = µ(d ) and F (x ) as in (8.7), we see that (8.4) implies (8.2). Proof Suppose that 1 ≤ U ≤ x /2. From (ii) and (iv) we see that ∑ x /(2U )<d ≤x /U ad F (x /d ) ≪ U (log U )c ∑ x /(2U )<d ≤x /U \|ad \|≪ x (log U )c . On taking U = 2 j and summing over j ≥ J we ﬁnd that ∑ d ≤x /2 J ad F (x /d ) ≪ x ∞∑ j = J 1 j c ≪c x J c−1 . This is small compared with x if J is large. Let A(x ) = ∑ d ≤x ad . T o treat the remaining range, x /2 J < d ≤ x , we sum by parts. W e do not use the Riemann– Stieltjes integral here because A( y) and F (x /y) may have common disconti- nuities. Let n0 = [x /2 J ] and n1 = [x ]. Then ∑ n0 <d ≤n1 ad F (x /d ) = ∑ n0 <d ≤n1 ( A(d ) − A(d − 1)) F (x /d ) = ∑ n0 <d ≤n1 A(d ) F (x /d ) − ∑ n0 −1<d ≤n1 −1 A(d ) F (x /(d + 1)) = A(n1 ) F (x /n1 ) − A(n0 ) F (x /(n0 + 1)) + ∑ n0 <d <n1 A(d ) ( F (x /d ) − F (x /(d + 1))) . Since A(ni ) = o(x ) and F (x /ni ) ≪J 1, the ﬁrst two terms are harmless. As the points x /d are monotonically arranged in the interval [1 ,2 J ], the sum above has absolute value not exceeding ( max d ≤x \| A(d )\| ) ∑ n0 <d <n1 \|F (x /d ) − F (x /(d + 1))\|≤ ( max d ≤x \| A(d )\| ) var[1,2 J ] F. By (i) and (iii) this is o(x ) for any given J . Thus the proof is complete. □	{ "creation_date": "2025-02-03", "file_name": "(Cambridge Studies in Advanced Mathematics 97) Hugh L. Montgomery, Robert C. Vaughan - Multiplicative number theory I_ Classical theory-Cambridge University Press (2006).pdf", "file_path": "/home/suryaremanan/Documents/IMO/augmentoolkit/original/input/(Cambridge Studies in Advanced Mathematics 97) Hugh L. Montgomery, Robert C. Vaughan - Multiplicative number theory I_ Classical theory-Cambridge University Press (2006).pdf", "file_size": 2365282, "file_type": "application/pdf", "last_modified_date": "2025-02-03", "page_label": "247" }
248 Further discussion of the Prime Number Theorem By means of a further application of Axer’s theorem, we now show that ∞∑ d =1 µ(d ) d = 0 (8.8) is also equivalent to the Prime Number Theorem. W e take ad = µ(d ) and F (x ) ={ x }= x − [x ] in Axer’s theorem. Thus from (8.4) we deduce that ∑ d ≤x µ(d ){x /d }= o(x ). But ∑ d ≤x µ(d )[x /d ] = 1 when x ≥ 1, so the left-hand side above is −1 + x ∑ d ≤x µ(d ) d . Since this is o(x ), we obtain (8.8). T o derive (8.4) from (8.8) is easier, in view of the following useful principle: Lemma 8.2If ∑ ∞ d=1 ad /d converges, then ∑ d ≤x ad = o(x ). Proof Let x be given, set r (u) = ∑ u<d ≤x ad /d , and note that ∑ d ≤x ad = ∫ x 0 r (u) du . But r (u) is bounded (independently of x ), and \|r (u)\| <ε for u > U0 , so the integral is ≪ U0 + εx . That is, the sum is o(x ), as desired. □ 8.1.1 Exercises 1. As in Section 2.2, let T (x ) = ∑ n≤x log n, and recall that T (x ) = x log x − x + O (log x ). (a) Show that T (x ) = ∑ d ≤x /Lambda1 (d )[x /d ]. (b) Show that x ∑ d ≤x /Lambda1 (d ) d = T (x ) − ∑ d ≤x {x /d }− ∑ d ≤x (/Lambda1 (d ) − 1){x /d }. (c) Use (8.2) and Axer’s theorem to show that the last sum above is o(x ). (d) Recall Exercise 2.1.1. (e) Show that (8.2) implies that ∑ d ≤x /Lambda1 (d ) d = log x − C0 + o(1), (8.9) and note how this compares with Theorem 2.7(a).	{ "creation_date": "2025-02-03", "file_name": "(Cambridge Studies in Advanced Mathematics 97) Hugh L. Montgomery, Robert C. Vaughan - Multiplicative number theory I_ Classical theory-Cambridge University Press (2006).pdf", "file_path": "/home/suryaremanan/Documents/IMO/augmentoolkit/original/input/(Cambridge Studies in Advanced Mathematics 97) Hugh L. Montgomery, Robert C. Vaughan - Multiplicative number theory I_ Classical theory-Cambridge University Press (2006).pdf", "file_size": 2365282, "file_type": "application/pdf", "last_modified_date": "2025-02-03", "page_label": "248" }
8.1 Relations equivalent to the Prime Number Theorem 249 (f) Apply Lemma 8.2 with ad = /Lambda1 (d ) − 1 to show that (8.9) implies (8.2). Hence (8.2) and (8.9) are equivalent. (g) Show that ∑ n≤x /Lambda1 (n){x /n}= (1 − C0 )x + o(x ). 2. (a) By recalling the proof of Theorem 2.2(c), or otherwise, show that (8.2) implies that ∫ x 1 ψ(u) u2 du = log x − 1 − C0 + o(1). (8.10) (b) Show that (8.10) implies (8.2). 3. Let b be deﬁned as in Theorem 2.7. (a) Imitate the proof of Theorem 2.7(d) to show that (8.2) implies that ∑ p≤x 1 p = log log x + b + o(1/log x ). (8.11) (b) Show that (8.11) implies (8.1). 4. (a) Use (8.10) and Exercise 5.2.12 to show that ∑ d ≤x µ(d ) d log(x /d ) = o(log x ). (8.12) (b) Show that (8.10) implies that ∑ d ≤x µ(d ) d log d = o(log x ). (8.13) (c) By partial summation, derive (8.4) from (8.13), and thus show that (8.2), (8.12) and (8.13) are all equivalent. (Note that a deeper assertion concerning the sum in (8.13) was already proved in Exercise 6.2.15.) 5. Let F (n) = ∑ d \|n f (d ) for all n. The opening remarks in Chapter 2 raise the possibility of a connection between the two relations (i) S(x ) = ∑ n≤x F (n) = cx + o(x ); (ii) ∑ ∞ d=1 f (d )/d = c. In Exercise 6.2.19 we have seen that (i) and the hypothesis f (n) ≪ 1 imply (ii). Apply Axer’s theorem with ad = f (d ), F (x ) ={ x } to show that (ii) and the hypothesis ∑ n≤x \| f (n)\|≪ x imply (i). 6. Let dk (n)b et h e kth divisor function, as deﬁned in Exercise 2.1.18. Put D0 (x ) = 1, and for positive integral k let Dk (x ) = ∑ n≤x dk (n). (a) Show that if k is a positive integer, then ∑ d ≤x µ(d ) Dk (x /d ) = Dk−1 (x ).	{ "creation_date": "2025-02-03", "file_name": "(Cambridge Studies in Advanced Mathematics 97) Hugh L. Montgomery, Robert C. Vaughan - Multiplicative number theory I_ Classical theory-Cambridge University Press (2006).pdf", "file_path": "/home/suryaremanan/Documents/IMO/augmentoolkit/original/input/(Cambridge Studies in Advanced Mathematics 97) Hugh L. Montgomery, Robert C. Vaughan - Multiplicative number theory I_ Classical theory-Cambridge University Press (2006).pdf", "file_size": 2365282, "file_type": "application/pdf", "last_modified_date": "2025-02-03", "page_label": "249" }
250 Further discussion of the Prime Number Theorem (b) Let g(n) be an arithmetic function, put G (x ) = ∑ n≤x g(n), and suppose that G (x ) = xP (log x ) + O (x /(log x )c ) where c > 1 and P is a polynomial of degree K . Let Pk be the polynomial deﬁned in Exercise 2.1.18, and explain why there exist constants ak so that P (z) = ∑ K +1 k=1 ak Pk (z). By applying Axer’s theorem with F (x ) = G (x ) −∑ K +1 k=1 ak Dk (x ), show that ∑ d ≤x µ(d )G (x /d ) = xQ (log x ) + o(x ) where Q is a polynomial of degree K − 1 with leading coefﬁcient equal to K times the leading coefﬁcient of P . 7. Show that Axer’s theorem holds with hypothesis (iv) replaced by the weaker condition that \|F (x )\|≤ ω(x )x for some non-negative function ω(x ) satisfy- ing ω(x ) ց and ∫∞ 1 ω(x )/xd x < ∞. 8.2 An elementary proof of the Prime Number Theorem As we saw in Exercise 2.1.5, a version of M ¨ obius inversion asserts that the two relationships B (x ) = ∑ n≤x A(x /n), A(x ) = ∑ n≤x µ(n) B (x /n) (8.14) are equivalent. Some familiar – and useful – examples of this pairing are displayed in T able 8.1. In many instances of (8.14), the functionsA(x ) and B (x ) are summatory functions of arithmetic functions a(n) and b(n), respec- tively , in which case a(n) and b(n) are linked by the more common M ¨ obius inversion b(n) = ∑ d \|n a(d ), a(n) = ∑ d \|n µ(d )b(n/d ). (8.15) The linear operator that takes A(x )t o B (x ) is continuous, but the transformation is nevertheless quite unstable. For example, the choice of the functions A(x )i n the second and third lines of T able 8.1 are very close, and yet the corresponding functionsB (x ) differ quite substantially . When the asymptotic rate of growth of A(x ) is known, it is easy to deduce that of B (x ), as a form of Abelian theorem. For example, if A(x ) ∼ x as x →∞ , then B (x ) ∼ x log x . However, from the fourth line of T able 8.1 we see that	{ "creation_date": "2025-02-03", "file_name": "(Cambridge Studies in Advanced Mathematics 97) Hugh L. Montgomery, Robert C. Vaughan - Multiplicative number theory I_ Classical theory-Cambridge University Press (2006).pdf", "file_path": "/home/suryaremanan/Documents/IMO/augmentoolkit/original/input/(Cambridge Studies in Advanced Mathematics 97) Hugh L. Montgomery, Robert C. Vaughan - Multiplicative number theory I_ Classical theory-Cambridge University Press (2006).pdf", "file_size": 2365282, "file_type": "application/pdf", "last_modified_date": "2025-02-03", "page_label": "250" }
8.2 An elementary proof of the Prime Number Theorem 251 T able 8.1 A (x) B (x) 1[ x ] xx ∑ n≤x 1 n = x log x + C0 x + O (1) [x ] ∑ n≤x d (n) = x log x + (2C0 − 1)x + O ( x 1/2 ) ψ(x ) ∑ n≤x log n = x log x − x + O (log x ) x log xx ∑ n≤x log x /n n = 1 2 x (log x )2 + C1 x log x + C2 x + O (1) some sort of T auberian converse would be useful, for the purpose of proving the Prime Number Theorem. Unfortunately , it is difﬁcult to establish anything stronger than the trivial estimate A(x ) ≪ ∑ n≤x \|B (x /n)\|. (8.16) From this we see that if B (x ) ≪ 1, then A(x ) ≪ x . This is rather weak, since the same upper bound for A(x ) can be deduced from a weaker upper bound for B (x ): From (8.16) we see that B (x ) ≪ x α, 0 ≤ α< 1 =⇒ A(x ) ≪α x . (8.17) As a ﬁrst application of this, we take A(x ) = ψ(x ) − x + 1 + C0 . Then from lines 1, 2, and 4 of T able 8.1 we see that B (x ) ≪ log x , and by (8.17) it follows that A(x ) ≪ x . That is, ψ(x ) ≪ x , which is the upper bound portion of Cheby- shev’s estimate. T o achieve greater success we construct a prime number sum in which the main term is larger thanO (x ). Theorem 8.3 (Selberg) Let /Lambda1 2 (n) = /Lambda1 (n) log n + ∑ bc=n /Lambda1 (b)/Lambda1 (c). Then for x ≥ 1, ∑ n≤x /Lambda1 2 (n) = 2x log x + O (x ). Clearly /Lambda1 2 (n) > 0 only when ω(n) ≤ 2. Thus the sum on the left above is analogous to ψ(x ) but with prime powers replaced by products of two prime powers, counted with suitable weights.	{ "creation_date": "2025-02-03", "file_name": "(Cambridge Studies in Advanced Mathematics 97) Hugh L. Montgomery, Robert C. Vaughan - Multiplicative number theory I_ Classical theory-Cambridge University Press (2006).pdf", "file_path": "/home/suryaremanan/Documents/IMO/augmentoolkit/original/input/(Cambridge Studies in Advanced Mathematics 97) Hugh L. Montgomery, Robert C. Vaughan - Multiplicative number theory I_ Classical theory-Cambridge University Press (2006).pdf", "file_size": 2365282, "file_type": "application/pdf", "last_modified_date": "2025-02-03", "page_label": "251" }
252 Further discussion of the Prime Number Theorem Proof W e begin by noting that ∑ d \|n /Lambda1 2 (d ) = ∑ d \|n /Lambda1 (d ) log d + ∑ d \|n ∑ bc=d /Lambda1 (b)/Lambda1 (c) = ∑ d \|n /Lambda1 (d ) log d + ∑ b\|n /Lambda1 (b) ∑ c\|n/b /Lambda1 (c). Here the sum over c is log n/b, so the above is = log n ∑ d \|n /Lambda1 (d ) = (log n)2 . (8.18) Hence by M ¨ obius inversion it follows that /Lambda1 2 (n) = ∑ d \|n µ(d )(log n/d )2 . (8.19) T ake now A(x ) = ∑ n≤x /Lambda1 2 (n) − 2x log x + c1 x + c2 (8.20) where c1 and c2 are constants to be chosen later. Then by (8.18) and lines 1, 2, and 5 of T able 8.1 we see that the corresponding B (x ) given by (8.14) is B (x ) = ∑ n≤x (log n)2 − 2x ∑ n≤x log x /n n + c1 x ∑ n≤x 1 n + c2 [x ]. By the integral test the ﬁrst sum is ∫x 1 (log u)2 du + O ((log x )2 ) = x (log x )2 − 2x log x + 2x + O ((log x )2 ). Hence the above is =− 2x log x + 2x − 2C1 x log x − 2C2 x + c1 x log x + c1 C0 x + c2 x + O ((log x )2 ). W e now choose c1 and c2 so that the leading terms cancel. That is, we take c1 = 2 + 2C1 and c2 =− 2 + 2C2 − c1 C0 . Then B (x ) ≪ (log x )2 , and hence by (8.17) it follows that A(x ) ≪ x . The desired estimate then follows from (8.20). □ Selberg’s identity may be modiﬁed in a variety of ways. For example, we note that ∑ n≤x /Lambda1 (n) log n = ∫ x 1 log ud ψ(u) = ψ(x ) log x − ∫ x 1 ψ(u) u du .	{ "creation_date": "2025-02-03", "file_name": "(Cambridge Studies in Advanced Mathematics 97) Hugh L. Montgomery, Robert C. Vaughan - Multiplicative number theory I_ Classical theory-Cambridge University Press (2006).pdf", "file_path": "/home/suryaremanan/Documents/IMO/augmentoolkit/original/input/(Cambridge Studies in Advanced Mathematics 97) Hugh L. Montgomery, Robert C. Vaughan - Multiplicative number theory I_ Classical theory-Cambridge University Press (2006).pdf", "file_size": 2365282, "file_type": "application/pdf", "last_modified_date": "2025-02-03", "page_label": "252" }
8.2 An elementary proof of the Prime Number Theorem 253 By Chebyshev’s estimate this last integral is ≪ x , and hence the above is = ψ(x ) log x + O (x ). (8.21) On inserting this in Selberg’s identity , we ﬁnd that ψ(x ) log x + ∑ n≤x ψ(x /n)/Lambda1 (n) = 2x log x + O (x ). (8.22) Our object is to show that each term on the left above is ∼ x log x as x → ∞. Suppose, to the contrary , that ψ(x ) is somewhat larger than anticipated, say ψ(x ) = ax with a > 1. By combining Mertens’ estimate ∑ n≤x /Lambda1 (n)/n = log x + O (1) with (8.22), we see that ψ( y)/y is on average approximately 2 − a as y runs over the points x /pk , counted with the appropriate weights. Note that 2 − a < 1. That is, if x is chosen so that ψ(x ) is unusually large, then ψ(x /pk ) must be unusually small for many prime powers pk . Such an argument may be repeated, so that one ﬁnds that ψ(x /( pk q ℓ)) is unusually large for many prime powers q ℓ. The points x /pk and x /( pk q ℓ) are highly interlacing, so that ψ( y) would have to switch rapidly back and forth between large and small values. However, ψ(x ) is a (weakly) increasing function, which implies that if it is unusually large at one point, then it continues to be unusually large for some time after. More precisely , ifψ(x ) ≥ ax with a > 1, then ψ( y) ≥ √ ay uniformly for x ≤ y ≤ √ax . Similarly , if ψ(x ) ≤ bx with b < 1 then ψ( y) ≤√ by uniformly for √ bx ≤ y ≤ x . Of course an interval on which ψ( y)i s large cannot overlap with one on which ψ( y) is small. One expects to reach a contradiction by showing that these intervals are too numerous and too long to all ﬁt in the interval [1,x ]. Our remaining task is to convert this intuitive line of reasoning into a rigorous proof. Let R(x ) be deﬁned by the relation ψ(x ) = x + R(x ). By combining the estimate of Mertens cited above with (8.22) we see that R(x ) log x + ∑ n≤x R(x /n)/Lambda1 (n) ≪ x . (8.23) Here the sum is a weighted average of values of R, but the total amount of weight, ∑ n≤x /Lambda1 (n) = ψ(x ), remains in doubt. T o overcome this difﬁculty , we iterate the identity (8.23) as follows: By replacing x in (8.23) by x /m we ﬁnd that R(x /m) log x /m + ∑ n≤x /m R(x /(mn ))/Lambda1 (n) ≪ x /m. W e multiply this by /Lambda1 (m) and sum over all m ≤ x , and thus ﬁnd that ∑ m≤x R(x /m)/Lambda1 (m) log x /m + ∑ mn ≤x R(x /(mn ))/Lambda1 (m)/Lambda1 (n) ≪ x log x .	{ "creation_date": "2025-02-03", "file_name": "(Cambridge Studies in Advanced Mathematics 97) Hugh L. Montgomery, Robert C. Vaughan - Multiplicative number theory I_ Classical theory-Cambridge University Press (2006).pdf", "file_path": "/home/suryaremanan/Documents/IMO/augmentoolkit/original/input/(Cambridge Studies in Advanced Mathematics 97) Hugh L. Montgomery, Robert C. Vaughan - Multiplicative number theory I_ Classical theory-Cambridge University Press (2006).pdf", "file_size": 2365282, "file_type": "application/pdf", "last_modified_date": "2025-02-03", "page_label": "253" }
254 Further discussion of the Prime Number Theorem W e multiply both sides of (8.23) by log x and subtract the above to see that R(x )(log x )2 =− ∑ n≤x R(x /n)/Lambda1 (n) log n + ∑ mn ≤x R(x /(mn ))/Lambda1 (m)/Lambda1 (n) + O (x log x ). (8.24) This has the advantage over (8.23) that we know how much weight resides in the coefﬁcients on the right-hand side, by virtue of Theorem 8.3. W e now formulate a T auberian principle that is appropriate to estimate the above expression. Lemma 8.4Suppose that a n ≥ 0 and b n ≥ 0 for all n, and that 1 2 x log x ≤ ∑ n≤x an ≤ 3 2 x log x , (8.25) 1 2 x log x ≤ ∑ n≤x bn ≤ 3 2 x log x (8.26) for all large x . Suppose also that ∑ n≤x an + bn ∼ 2x log x (8.27) as x →∞ . Finally, suppose that r (u) is a function such that \|r (u)\|≤ βu (8.28) for all large u where 0 <β ≤ 1, and that r (v) − r (u) ≥− (v − u) (8.29) when v ≥ u. Then ⏐ ⏐ ⏐ ⏐ ∑ n≤x (an − bn )r (x /n) ⏐ ⏐ ⏐ ⏐≤ ( β − β2 100 + o(1) ) x (log x )2 . Proof Without loss of generality the hypotheses hold for all x ≥ 1, u ≥ 1, since changes in the deﬁnitions of an ,bn for small n, and r (u) for small u entail additional error terms of magnitude O (x log x ). It sufﬁces to show that ∑ n≤x (an − bn )r (x /n) ≤ ( β − β2 100 + o(1) ) x (log x )2 , (8.30) since the reverse inequality can then be derived by exchanging the roles of an and bn . By applying ﬁrst (8.28) and then (8.27) we see that the left-hand side above is trivially ≤ βx ∑ n≤x an + bn n ∼ βx (log x )2 . (8.31)	{ "creation_date": "2025-02-03", "file_name": "(Cambridge Studies in Advanced Mathematics 97) Hugh L. Montgomery, Robert C. Vaughan - Multiplicative number theory I_ Classical theory-Cambridge University Press (2006).pdf", "file_path": "/home/suryaremanan/Documents/IMO/augmentoolkit/original/input/(Cambridge Studies in Advanced Mathematics 97) Hugh L. Montgomery, Robert C. Vaughan - Multiplicative number theory I_ Classical theory-Cambridge University Press (2006).pdf", "file_size": 2365282, "file_type": "application/pdf", "last_modified_date": "2025-02-03", "page_label": "254" }
8.2 An elementary proof of the Prime Number Theorem 255 W e write the left-hand side of (8.30) in the form βx ∑ n≤x an + bn n − ∑ n≤x an (βx n − r (x /n) ) − ∑ n≤x bn (βx n + r (x /n) ) . By (8.31), this is ∼ βx (log x )2 − SA − SB , say . Note that both factors of the summands in SA are non-negative, so that SA ≥ 0. Similarly , SB ≥ 0. W e need to show that SA + SB ≥ (β2 100 + o(1) ) x (log x )2 . (8.32) T o this end we show that ∑ y<n≤16 y an (βx n − r (x /n) ) + bn (βx n + r (x /n) ) ≥ 1 16 β2 x log y (8.33) for all large y. Then (8.32) follows on summing this over y = x 16−k ,1 ≤ k ≤ [(log x )/log 16] . In proving (8.33) we consider three cases. Case 1. r (u) ≤ 1 2 βu for all u ∈ [ x 16 y , x 4 y ]. Then r (x /n) ≤ 1 2 βx /n for all n ∈ [4 y,16 y], and hence ∑ y<n≤16 y an (βx n − r (x /n) ) ≥ 1 2 βx ∑ 4 y<n≤16 y an n . Since the denominator does not exceed 16 y, the above is ≥ βx 32 y ∑ 4 y<n≤16 y an . Here the sum is ∑ n≤16 y an − ∑ n≤4 y an , which by (8.25) is ≥ 8 y log 16 y − 6 y log 4 y > 2 y log y. Thus the above is ≥ βx 16 log y. Since β ≤ 1, this gives (8.33) in this case. Case 2. r (u) ≥− 1 2 βu for all u ∈ [ x 4 y , x y ]. Then r (x /n) ≥− 1 2 βx /n for n ∈ [ y,4 y]. Arguing as in the preceding case, but using (8.26) instead of (8.25), we ﬁnd that ∑ y<n≤4 y bn (βx n + r (x /n) ) ≥ 1 2 βx ∑ y<n≤4 y bn n ≥ βx 8 y ∑ y<n≤4 y bn ≥ βx log y 16 . This gives (8.33) in this case. If neither Case 1 nor Case 2 applies, then we have	{ "creation_date": "2025-02-03", "file_name": "(Cambridge Studies in Advanced Mathematics 97) Hugh L. Montgomery, Robert C. Vaughan - Multiplicative number theory I_ Classical theory-Cambridge University Press (2006).pdf", "file_path": "/home/suryaremanan/Documents/IMO/augmentoolkit/original/input/(Cambridge Studies in Advanced Mathematics 97) Hugh L. Montgomery, Robert C. Vaughan - Multiplicative number theory I_ Classical theory-Cambridge University Press (2006).pdf", "file_size": 2365282, "file_type": "application/pdf", "last_modified_date": "2025-02-03", "page_label": "255" }
256 Further discussion of the Prime Number Theorem Case 3. There is a u 1 ∈ [ x 16 y , x 4 y ] such that r (u1 ) ≥ 1 2 βu1 , a n dau 2 ∈ [ x 4 y , x y ] such that r (u2 ) ≤− 1 2 βu2 . Let u4 be the inf of those u ≥ u1 such that r (u) ≤− 1 2 βu. W e show that r (u4 ) =− 1 2 βu4 . Suppose that r (u4 ) > − 1 2 βu4 , say r (u4 ) + 1 2 βu4 = δ> 0. Suppose that u4 ≤ v< u4 + δ 1 − 1 2 β. (8.34) Then by (8.29) we see that r (v) ≥ r (u4 ) − (v − u4 ) =− 1 2 βu4 + δ − (v − u4 ). From the upper bound in (8.34) we deduce that the above expression is > − 1 2 βv. That is, the inequality r (u) ≤− 1 2 βu holds at no point of the interval (8.34). Since this contradicts the deﬁnition of u4 , it follows that r (u4 ) ≤− 1 2 βu4 . Now suppose that r (u4 ) < − 1 2 βu4 , say −r (u4 ) − 1 2 βu4 = δ> 0. Suppose also that u4 − δ 1 − 1 2 β ≤ u ≤ u4 . (8.35) Then by (8.29) we see that r (u) ≤ r (u4 ) + (u4 − u) =− 1 2 βu4 − δ + (u4 − u). From the lower bound in (8.35) we deduce that this expression is ≤− 1 2 βu. That is, the inequality r (u) ≤− 1 2 βu holds throughout the interval (8.35). Since this contradicts the deﬁnition of u4 , we conclude that r (u4 ) =− 1 2 βu4 . Put u3 = 1 − 1 2 β 1 + 1 2 βu4 , and suppose that u3 < u ≤ u4 . (8.36) Then by (8.29) we see that r (u) ≤ r (u4 ) + (u4 − u) =− 1 2 βu4 + (u4 − u). From the lower bound in (8.36) we deduce that this expression is < 1 2 βu. That is, the inequality r (u) ≥ 1 2 βu holds at no point of the interval (8.36), and hence u1 ≤ u3 .	{ "creation_date": "2025-02-03", "file_name": "(Cambridge Studies in Advanced Mathematics 97) Hugh L. Montgomery, Robert C. Vaughan - Multiplicative number theory I_ Classical theory-Cambridge University Press (2006).pdf", "file_path": "/home/suryaremanan/Documents/IMO/augmentoolkit/original/input/(Cambridge Studies in Advanced Mathematics 97) Hugh L. Montgomery, Robert C. Vaughan - Multiplicative number theory I_ Classical theory-Cambridge University Press (2006).pdf", "file_size": 2365282, "file_type": "application/pdf", "last_modified_date": "2025-02-03", "page_label": "256" }
8.2 An elementary proof of the Prime Number Theorem 257 T o summarize, we have x 16 y ≤ u1 ≤ u3 ≤ u4 ≤ x y and \|r (u)\|≤ 1 2 βu for u3 < u ≤ u4 . Hence ∑ x /u4 ≤n<x /u3 an (βx n − r (x /n) ) + bn (βx n + r (x /n) ) ≥ 1 2 βx ∑ x /u4 ≤n<x /u3 an + bn n = (1 2 β + o(1) ) x ( (log x /u3 )2 − (log x /u4 )2 ) . (8.37) T o estimate the last factor above we note that log x u3 − log x u4 = log 1 + 1 2 β 1 − 1 2 β = ∞∑ r =0 β2r +1 (2r + 1)22r >β. Also, since u3 and u4 do not exceed x /y, it follows that log x u3 + log x u4 ≥ 2 log y. Hence the expression (8.37) is ≥ ( β2 + o(1) ) x log y. Thus we have (8.33) in this case also, and the proof of Lemma 8.4 is complete. □ T o complete the proof of the Prime Number Theorem we apply Lemma 8.4 with an = /Lambda1 (n) log n, bn = ∑ bc=n /Lambda1 (b)/Lambda1 (c). W e combine Chebyshev’s estimates in the form (log 2 + o(1))x ≤ ψ(x ) ≤ (2 log 2 + o(1))x with (8.21) to see that (log 2 + o(1))x log x ≤ ∑ n≤x an ≤ (2 log 2 + o(1))x log x . (8.38) This gives (8.25), and (8.27) is Selberg’s identity as expressed in Theorem 8.3. T o obtain (8.26) it sufﬁces to subtract (8.38) from (8.27). W e apply the lemma withr (u) = R(u) = ψ(u) − u. Then r (v) − r (u) = ∑ u<n≤v /Lambda1 (n) − (v − u) ≥− (v − u), so we have (8.28). Let α = lim sup \|r (u)\|/u. Our object is to show that α = 0. W e know that α ≤ 1/2, by Chebyshev’s estimates. Suppose that α> 0, and	{ "creation_date": "2025-02-03", "file_name": "(Cambridge Studies in Advanced Mathematics 97) Hugh L. Montgomery, Robert C. Vaughan - Multiplicative number theory I_ Classical theory-Cambridge University Press (2006).pdf", "file_path": "/home/suryaremanan/Documents/IMO/augmentoolkit/original/input/(Cambridge Studies in Advanced Mathematics 97) Hugh L. Montgomery, Robert C. Vaughan - Multiplicative number theory I_ Classical theory-Cambridge University Press (2006).pdf", "file_size": 2365282, "file_type": "application/pdf", "last_modified_date": "2025-02-03", "page_label": "257" }
258 Further discussion of the Prime Number Theorem choose β,0 <β ≤ 1 so that β − β2 100 <α<β. By combining the conclusion of Lemma 8.4 with (8.24) we deduce that α ≤ β − β2 /100, a contraction. Thus α = 0, and the proof of the Prime Number Theorem is complete. 8.2.1 Exercises 1. For which entries in T able 8.1 are A(x ) and B (x ) summatory functions of arithemtic functions a(n) and b(n) related as in (8.15) ? 2. If A(x ) = M (x ): = ∑ n≤x µ(n) in (8.14), then what is the function B (x )? 3. (a) V erify the Dirichlet series identity (ζ′ ζ (s) )′ + (ζ′ ζ (s) )2 = ζ′′ ζ (s). (b) Compute the Dirichlet series coefﬁcients of the three functions in the above identity , and thus give a proof of (8.18) by means of formal Dirich- let series. (c) Compute the leading term of the Laurent expansions of the three func- tions above, at the point s = 1. (d) Suppose that ρ is a zero of ζ(s) of multiplicity m > 0. Compute the singular portion of the Laurent expansions of the three functions above, ats = ρ. Note that the pole of ζ′′/ζ at s = ρ is simple if and only if ρ is a simple zero of ζ(s). 4. Let a = lim sup x →∞ ψ(x )/x and b = lim inf x →∞ ψ(x )/x . Suppose that a sequence xν tending to inﬁnity is chosen so that lim ν→∞ ψ(xν)/xν = a. Use (8.22) to show that for each νa prime pν can be selected so that xν/pν →∞ and lim inf ν→∞ ψ(xν/pν)/(xν/pν) ≤ 2 − a. Thus show that a + b ≤ 2. By a similar argument, show that a + b ≥ 2. Hence demonstrate that the relation a + b = 2 is a consequence of (8.22). 5. (a) Show that log x ∑ pk ≤x k≥2 log p + ∑ pk q ℓ≤x k+ℓ≥3 (log p) log q ≪ x . Here p and q denote prime numbers. (b) As usual, let ϑ(x ) = ∑ p≤x log p, and use Selberg’s identity to show that ϑ(x ) log x + ∑ p≤x ϑ(x /p) log p = 2x log x + O (x ).	{ "creation_date": "2025-02-03", "file_name": "(Cambridge Studies in Advanced Mathematics 97) Hugh L. Montgomery, Robert C. Vaughan - Multiplicative number theory I_ Classical theory-Cambridge University Press (2006).pdf", "file_path": "/home/suryaremanan/Documents/IMO/augmentoolkit/original/input/(Cambridge Studies in Advanced Mathematics 97) Hugh L. Montgomery, Robert C. Vaughan - Multiplicative number theory I_ Classical theory-Cambridge University Press (2006).pdf", "file_size": 2365282, "file_type": "application/pdf", "last_modified_date": "2025-02-03", "page_label": "258" }
8.3 The Wiener–Ikehara T auberian theorem 259 6. Show that ∑ d \|n µ(d )(log n/d )2 = /Lambda1 (n) log n + ∑ d \|n /Lambda1 (d )/Lambda1 (n/d ). 7. Let k be a positive integer, and put /Lambda1 k (n) = ∑ d \|n µ(d )(log n/d )k . (a) Show that /Lambda1 k+1 (n) = /Lambda1 k (n) log n + ∑ d \|n /Lambda1 k (d )/Lambda1 (n/d ). (b) Show that /Lambda1 k (n) ≥ 0 for all n, and that if /Lambda1 k (n) > 0, then ω(n) ≤ k. 8. Let c and M be positive constants, and suppose that f (x ) is a function deﬁned on [1 ,∞) such that (i) \| ∫x 1 f (u)u−2 du \|≤ M for all x ≥ 1, and also (ii) \| f (u) − f (v)\|≤ c\|u − v\| whenever u ≥ 1 and v ≥ 1. Put α = lim sup x →∞ \| f (x )\| x ,β = lim sup x →∞ 1 log x ∫ x 1 \| f (u)\| u2 du . Show that β ≤ α(1 − α2 /(32cM )). 8.3 The Wiener–Ikehara T auberian theorem In Chapter 6 we developed some understanding of the analytic behaviour of the zeta function, which allowed us to show thatζ(s) ̸=0 for σ ≥ 1 − c/log τ, which in turn permitted us to establish the Prime Number Theorem with an error term≪ x exp(−c√ log x ). On the other hand, it is reasonable to ask what is the least information concerning the zeta function that would sufﬁce to establish the Prime Number Theorem in the weak form (8.1). In this section we establish a general T auberian theorem, from which the Prime Number Theorem follows from the information that the functions ζ(s) − 1 s − 1 ,ζ ′(s) + 1 (s − 1)2 are continuous in the closed half-plane σ ≥ 1, and that ζ(1 + it ) ̸=0 (8.39) for all real t . Conversely from (8.2) we see that − ζ′ ζ (s) = s s − 1 + s ∫ ∞ 1 ψ(x ) − x x s+1 dx = o (1 σ − 1 ) as σ → 1+ with t ﬁxed, t ̸=0. But if ζ(s) had a zero of multiplicity m at 1 + it , then ζ′ ζ (s) ∼ m s − 1	{ "creation_date": "2025-02-03", "file_name": "(Cambridge Studies in Advanced Mathematics 97) Hugh L. Montgomery, Robert C. Vaughan - Multiplicative number theory I_ Classical theory-Cambridge University Press (2006).pdf", "file_path": "/home/suryaremanan/Documents/IMO/augmentoolkit/original/input/(Cambridge Studies in Advanced Mathematics 97) Hugh L. Montgomery, Robert C. Vaughan - Multiplicative number theory I_ Classical theory-Cambridge University Press (2006).pdf", "file_size": 2365282, "file_type": "application/pdf", "last_modified_date": "2025-02-03", "page_label": "259" }
260 Further discussion of the Prime Number Theorem when s is near 1 + it . Since this is possible only when m = 0, we have (8.39). The above observations can be paraphrased as ‘the Prime Number Theorem is equivalent to the assertion (8.39)’, although one needs to bear in mind the continuity conditions also. Suppose that α(s) = ∑ ∞ n=1 an n−s . In Section 5.2 we derived information concerning partial sums of this series at s = 1 from the behaviour of α(σ)a s σ → 1+. W e now take much stronger hypotheses that concern α(s) throughout the closed half-plane σ ≥ 1, but we obtain from them much stronger conclu- sions, concerning partial sums of the series at s = 0. Our proof of the Hardy– Littlewood T auberian theorem (Theorem 5.7) depended on a simple lemma con- cerning one-sided polynomial approximation (Lemma 5.8). Our new approach depends similarly on a corresponding lemma concerning one-sided trigonomet- ric approximation, as follows. Lemma 8.5Let E (x ) = ex for x ≤ 0, and E (x ) = 0 for x > 0. F or any given ε> 0 there is a T and continuous functions f +(x ),f −(x ) with f ± ∈ L 1 (R) such that (i) f−(x ) ≤ E (x ) ≤ f+(x ) for all real x ; (ii) ˆf ±(t ) = 0 for \|t \|≥ T ; (iii) ∫∞ −∞ f+(x ) dx < 1 + ε, ∫∞ −∞ f−(x ) dx > 1 − ε. Before proving the above, we ﬁrst explore its consequences. Since thef± ∈ L 1 (R), it follows that the Fourier transforms ˆf ±(t ) are uni- formly continuous. Thus from (ii) above it follows that ˆf ±(±T ) = 0, so that ˆf ±(t ) = 0 for all t with \|t \|≥ T . Since the f± are also continuous, it follows by the Fourier integral theorem that lim τ→∞ ∫ τ −τ (1 −\| t \|/τ) ˆf ±(t )e(tx ) dt = f±(x ) for all x . But the functions ˆf ± are supported on the ﬁxed interval [ −T ,T ], so the limit on the left above is simply ∫T −T ˆf ±(t )e(tx ) dt . That is, f±(x ) = ∫ T −T ˆf ±(t )e(tx ) dt (8.40) for all x . It may be further noted that ∫T −T ˆf ±(t )e2πitz dt is an entire function of z. Thus f±(x ) is the restriction to the real axis of an entire function. Theorem 8.6 (Wiener–Ikehara) Suppose that the function a (u) is non- negative and increasing on [0,∞), that α(s) = ∫ ∞ 0 e−us da (u)	{ "creation_date": "2025-02-03", "file_name": "(Cambridge Studies in Advanced Mathematics 97) Hugh L. Montgomery, Robert C. Vaughan - Multiplicative number theory I_ Classical theory-Cambridge University Press (2006).pdf", "file_path": "/home/suryaremanan/Documents/IMO/augmentoolkit/original/input/(Cambridge Studies in Advanced Mathematics 97) Hugh L. Montgomery, Robert C. Vaughan - Multiplicative number theory I_ Classical theory-Cambridge University Press (2006).pdf", "file_size": 2365282, "file_type": "application/pdf", "last_modified_date": "2025-02-03", "page_label": "260" }
8.3 The Wiener–Ikehara T auberian theorem 261 converges for all s with σ> 1, and that r (s): = α(s) − c/(s − 1) extends to a continuous function in the closed half-plane σ ≥ 1. Then ∫ x 0 1 da (u) = cex + o(ex ) as x →∞ . By making the change of variable a(u) = A (eu ), we obtain the following equivalent formulation. Corollary 8.7(Wiener–Ikehara) Suppose that A (v) is non-negative and in- creasing on [1,∞), that α(s) = ∫ ∞ 1 v−s dA (v) converges for all s with σ> 1, and that r (s): = α(s) − c/(s − 1) extends to a continuous function in the closed half-plane σ ≥ 1. Then ∫ x 1 1 dA (v) = cx + o(x ) as x →∞ . By setting A(v) = ∑ n<v an we obtain a useful T auberian theorem for Dirich- let series. Corollary 8.8(Wiener–Ikehara) Suppose that a n ≥ 0 for all n, that α(s) = ∞∑ n=1 an n−s converges for all s with σ> 1, and that r (s): = α(s) − c/(s − 1) extends to a continuous function in the closed half-plane σ ≥ 1. Then ∑ n≤x an = cx + o(x ) as x →∞ . By taking an = /Lambda1 (n), we see that (8.39) gives the hypotheses with c = 1, and hence we obtain the Prime Number Theorem in the form (8.2). Proof of Theorem 8.6Ta k eδ> 0, and let E (u) be as in Lemma 8.5. Then ∫ x 0 e−δu da (u) = ex ∫ ∞ 0 E (u − x )e−(1+δ)u da (u), which by Lemma 8.5(i) is ≤ ex ∫ ∞ 0 f+(u − x )e−(1+δ)u da (u).	{ "creation_date": "2025-02-03", "file_name": "(Cambridge Studies in Advanced Mathematics 97) Hugh L. Montgomery, Robert C. Vaughan - Multiplicative number theory I_ Classical theory-Cambridge University Press (2006).pdf", "file_path": "/home/suryaremanan/Documents/IMO/augmentoolkit/original/input/(Cambridge Studies in Advanced Mathematics 97) Hugh L. Montgomery, Robert C. Vaughan - Multiplicative number theory I_ Classical theory-Cambridge University Press (2006).pdf", "file_size": 2365282, "file_type": "application/pdf", "last_modified_date": "2025-02-03", "page_label": "261" }
262 Further discussion of the Prime Number Theorem By (8.40) this is = ex ∫ ∞ 0 ∫ T −T ˆf +(t )e(tu − tx ) dt e −(1+δ)u da (u). By Fubini’s theorem we may interchange the order of integration. Thus the above is = ex ∫ T −T ˆf +(t )e(−tx ) ∫ ∞ 0 e−(1+δ−2πit )u da (u) dt = ex ∫ T −T ˆf +(t )e(−tx )α(1 + δ − 2πit ) dt . (8.41) If a(u) = eu , then α(s) = 1/(s − 1), and thus from the above calculation we see in particular that ∫ ∞ 0 f+(u − x )e−δu du = ∫ T −T ˆf +(t )e(−tx ) 1 δ − 2πit dt . On multiplying both sides by cex and combining this with (8.41), we deduce that ∫ x 0 e−δu da (u) ≤ ex ∫ T −T ˆf +(t )e(−tx )r (1 + δ − 2πit ) dt + cex ∫ ∞ 0 f+(u − x )e−δu du . Since r (s) is uniformly continuous in the closed rectangle 1 ≤ σ ≤ 1 + δ, \|t \|≤ 2πT , each of the above three terms tends to a limit as δ → 0+. Thus ∫ x 0 1 da (u) ≤ ex ∫ T −T ˆf +(t )e(−tx ) r (1 − 2πit ) dt + cex ∫ ∞ 0 f+(u − x ) du . W e divide through by ex and let x tend to inﬁnity . The ﬁrst integral on the right tends to 0 by the Riemann–Lebesgue lemma, and the second integral on the right tends to ∫∞ −∞ f+(u) du . Thus we see that lim sup x →∞ e−x ∫ x 0 1 da (u) ≤ c ∫ ∞ −∞ f+(u) du ≤ c(1 + ε) by Lemma 8.5(iii). By using f− similarly we may also show that lim inf x →∞ e−x ∫ x 0 1 da (u) ≥ c(1 − ε). Since ε may be taken arbitrarily small, we obtain the stated result, apart from the need to prove Lemma 8.5. □	{ "creation_date": "2025-02-03", "file_name": "(Cambridge Studies in Advanced Mathematics 97) Hugh L. Montgomery, Robert C. Vaughan - Multiplicative number theory I_ Classical theory-Cambridge University Press (2006).pdf", "file_path": "/home/suryaremanan/Documents/IMO/augmentoolkit/original/input/(Cambridge Studies in Advanced Mathematics 97) Hugh L. Montgomery, Robert C. Vaughan - Multiplicative number theory I_ Classical theory-Cambridge University Press (2006).pdf", "file_size": 2365282, "file_type": "application/pdf", "last_modified_date": "2025-02-03", "page_label": "262" }
8.3 The Wiener–Ikehara T auberian theorem 263 Proof of Lemma 8.5 W e assume, as we may , that T ≥ 1. Let /Delta1 T (x ) = T (sin πTx πTx )2 , JT (x ) = 3T 4 (sin πTx /2 πTx /2 )4 be the Fej´ er and Jackson kernels, respectively . These functions have a peak of height≍T and width ≍ 1/T at 0, and have total mass 1. Set f (x ) = ( E ⋆ JT )(x ) = ∫ ∞ −∞ E (u) JT (x − u) du . This is a weighted average of the values of E (u) with special emphasis on those u near x . W e show that f (x ) = E (x ) + O (min(1,1/(Tx )2 )). (8.42) T o establish this we consider several cases. If \|x \|≤ 1/T we simply observe that 0 ≤ f (x ) ≤ ∫∞ −∞ JT (u) du = 1. If x ≥ 1/T we observe that 0 ≤ f (x ) ≪ T −3 ∫0 −∞(x − u)−4 du ≪ 1/(Tx )3 . By the calculus of residues it is easy to show that ∫∞ −∞ JT (u) du = 1. Hence f (x ) − E (x ) = ∫ ∞ −∞ ( E (u) − E (x )) JT (x − u) du . Next, suppose that −1 ≤ x ≤− 1/T .I f2 x ≤ u ≤ 0, then E (u) − E (x ) = ex (eu−x − 1) = ex (u − x + O ((u − x )2 )). Thus ∫ 0 2x ( E (u) − E (x )) JT (x − u) du =− ex ∫ −x x uJ T (u) du + O (∫ −x x u2 JT (u) du ) . Here the ﬁrst integral on the right vanishes because the integrand is an odd function, and the second integral is≪ 1/T 2 . On the other hand, ∫ ∞ 0 ( E (u) − E (x )) JT (x − u) du ≪ T −3 ∫ ∞ −x u−4 du ≪ 1/\|Tx \|3 , and similarly ∫2x −∞ ≪ 1/\|Tx \|3 , so we have (8.42) in this case also. Finally , suppose that x ≤− 1. Then E (u) − E (x ) = ex (u − x + O ((u − x )2 )) for x − 1 ≤ u ≤ x + 1, so that ∫ x +1 x −1 ( E (u) − E (x )) JT (x − u) du =− ex ∫ 1 −1 uJ T (u) du + O ( ex ∫ 1 −1 u2 JT (u) du ) ≪ ex T −2 ,	{ "creation_date": "2025-02-03", "file_name": "(Cambridge Studies in Advanced Mathematics 97) Hugh L. Montgomery, Robert C. Vaughan - Multiplicative number theory I_ Classical theory-Cambridge University Press (2006).pdf", "file_path": "/home/suryaremanan/Documents/IMO/augmentoolkit/original/input/(Cambridge Studies in Advanced Mathematics 97) Hugh L. Montgomery, Robert C. Vaughan - Multiplicative number theory I_ Classical theory-Cambridge University Press (2006).pdf", "file_size": 2365282, "file_type": "application/pdf", "last_modified_date": "2025-02-03", "page_label": "263" }
264 Further discussion of the Prime Number Theorem which is ≪ 1/(Tx )2 . On the other hand, ∫ x −1 −∞ ( E (u) − E (x )) JT (x − u) du ≪ ex T −3 ∫ ∞ 1 u−4 du ≪ (Tx )−2 , and ∫ ∞ x +1 ( E (u) − E (x )) JT (x − u) du ≪ T −3 x −4 , so we again have (8.42). Clearly /Delta1 T (x ) ≪ T min(1,1/(Tx )2 ), but there is no inequality in the reverse direction because /Delta1 T (x ) vanishes at integral multiples of 1 /T . T o overcome this difﬁculty we consider also a translate of the Fej´ er kernel. Since /Delta1 T (x ) + /Delta1 T (x + 1/(2T )) ≫ T min(1,1/(Tx )2 ), we take f±(x ) = f (x ) ± C T (/Delta1 T (x ) + /Delta1 T (x + 1/(2T ))) . By (8.42) we see that if C is taken large enough, then f−(x ) ≤ E (x ) ≤ f+(x ) for all x . By Fubini’s theorem it is easy to see that if f1 , f2 ∈ L 1 (R) then the convo- lution f1 ⋆ f2 is also in L 1 (R), and also that ˆf1 ⋆ f2 (t ) = ˆf1 (t )ˆf2 (t ). Hence in particular, f ∈ L 1 (R) and ˆf (t ) = ˆE (t )ˆJT (t ). But ˆJT (t ) = 0 for \|t \|≥ T , so ˆf (t ) = 0 for \|t \|≥ T . Also, ˆ/Delta1 T (t ) = 0 for \|t \|≥ T , and we see that the functions f± have the property (ii). Finally , we note by Fubini’s theorem that ∫ ∞ −∞ f (x ) dx = (∫ ∞ −∞ E (x ) dx )(∫ ∞ −∞ JT (u) du ) = 1 · 1 = 1, and hence ∫∞ −∞ f±(x ) dx = 1 ± 2C /T . Thus we have (iii) if T ≥ C /ε, so the proof is complete. □ 8.3.1 Exercises 1. Use the Wiener–Ikehara theorem (Theorem 8.6) to show that M (x ) = o(x ). 2. (Dressler 1970; cf. Bateman 1972) Let f (n) denote the number of positive integers k such that ϕ(k) = n. (a) Show that if σ> 1, then ∞∑ n=1 f (n) ns = ∞∑ k=1 1 ϕ(k)s = ∏ p ( 1 + 1 ϕ( p)s + 1 ϕ( p2 )s +··· ) , and explain why this is not an Euler product in the usual sense.	{ "creation_date": "2025-02-03", "file_name": "(Cambridge Studies in Advanced Mathematics 97) Hugh L. Montgomery, Robert C. Vaughan - Multiplicative number theory I_ Classical theory-Cambridge University Press (2006).pdf", "file_path": "/home/suryaremanan/Documents/IMO/augmentoolkit/original/input/(Cambridge Studies in Advanced Mathematics 97) Hugh L. Montgomery, Robert C. Vaughan - Multiplicative number theory I_ Classical theory-Cambridge University Press (2006).pdf", "file_size": 2365282, "file_type": "application/pdf", "last_modified_date": "2025-02-03", "page_label": "264" }
8.3 The Wiener–Ikehara T auberian theorem 265 (b) Let the above Dirichlet series be F (s). Show that F (s) = ζ(s)G (s) for σ> 1, where G (s) = ∏ p ( 1 − 1 ps + 1 ( p − 1)s ) . (c) By writing 1 ( p − 1)s − 1 ps = s ∫ p p−1 u−s−1 du , show that the above is ≪ p−σ−1 for any ﬁxed s. (d) Let K be a compact set in the complex plane, and let σ0 = mins∈K σ. Show that ( p − 1)−s − p−s ≪ p−σ0 −1 uniformly for s ∈ K. (e) Show the product G (s) converges locally uniformly in the half-plane σ> 0, and hence represents an analytic function in this region. (f) Show that G (1) = ζ(2)ζ(3)/ζ(6). (g) Use the Wiener–Ikehara theorem (Theorem 8.6) to show that the number of integers k such that ϕ(k) ≤ x is asymptotic to G (1)x as x →∞ . 3. Show that Corollary 8.8 still holds if the hypothesis an ≥ 0 is replaced by the weaker hypothesis that there is a constant C such that an ≥ C for all n. 4. Let σs (n) = ∑ d \|n d s , and let cq (n) be Ramanujan’s sum, as discussed in Section 4.1. (a) Show that ifn is a positive integer, then ∞∑ q =1 cq (n) q s = σ1−s (n) ζ(s) (σ> 1). (b) Show that if n is a ﬁxed positive integer, then ∑ q ≤x cq (n) = o(x )a s x →∞ . (c) Show that if n is a positive integer, then ∑ q ≤x cq (n) [x q ] = ∑ d \|n d ≤x d . (d) By Axer’s theorem, or otherwise, show that if n is a positive integer, then ∞∑ q =1 cq (n) q = 0. (See also Exercise 4.1.8.) 5. (Graham & V aaler 1981) Let f+(x ) and f−(x ) be as in Lemma 8.5. (a) Use the Poisson summation formula to show that ∞∑ n=−∞ f+(n/T ) = T ∞∑ k=−∞ ˆf +(kT ) .	{ "creation_date": "2025-02-03", "file_name": "(Cambridge Studies in Advanced Mathematics 97) Hugh L. Montgomery, Robert C. Vaughan - Multiplicative number theory I_ Classical theory-Cambridge University Press (2006).pdf", "file_path": "/home/suryaremanan/Documents/IMO/augmentoolkit/original/input/(Cambridge Studies in Advanced Mathematics 97) Hugh L. Montgomery, Robert C. Vaughan - Multiplicative number theory I_ Classical theory-Cambridge University Press (2006).pdf", "file_size": 2365282, "file_type": "application/pdf", "last_modified_date": "2025-02-03", "page_label": "265" }
266 Further discussion of the Prime Number Theorem (b) Explain why the right-hand side above is = T ˆf +(0) = T ∫ R f+(x ) dx . (c) Explain why the left-hand side above is ≥ (1 − e−1/T )−1 . (d) Deduce that ∫ R f+(x ) dx ≥ 1 T (1 − e−1/T ) . (e) Suppose that T ≥ 2. Show that the right-hand side above is = 1 + 1/(2T ) + O (1/T 2 ). (f) Show similarly that ∫ R f−(x ) dx ≤ 1 T (e1/T − 1) , and that the right-hand side is = 1 − 1/(2T ) + O (1/T 2 ) when T ≥ 2. 8.4 Beurling’s generalized prime numbers One of the most valuable generalizations of the Prime Number Theorem is to algebraic number ﬁelds. Suppose thatK is an algebraic number ﬁeld of degree d over the rationals, and let OK denote the ring of algebraic integers in K .F o r some ﬁelds K the members of OK factor uniquely into primes, but in general this is not the case. However, it is always true that ideals in OK factor uniquely into prime ideals. For an ideal a of OK , let N (a) denote its norm, which is to say the size of the quotient ring OK /a.F o r σ> 1 we can deﬁne the Dedekind zeta function of K by the absolutely convergent series ζK (s) = ∑ a N (a)−s . This is an ordinary Dirichlet series, since the N (a) are positive integers, and thus the above can be written in the form ∑ an n−s where an is the number of ideals with norm n. Counting ideals a with N (a) ≤ x is rather like counting rational integers. The ideals can be parametrized by the points of a lattice in Rd , so one is counting lattice points in a certain region, which is approximately the volume of that region, and thus it can be shown that the numberI (x ) of ideals a with N (a) ≤ x is I (x ) = cx + O ( x 1−1/d ) (8.43) where c = c(K ) is a certain positive constant, called the ideal density . Here the implicit constant may also depend on K , which we assume is ﬁxed. By Theorem 1.3 it follows that ζK (s) = s ∫ ∞ 1 I (x )x −s−1 dx = cs s − 1 + s ∫ ∞ 1 ( I (x ) − cx )x −s−1 dx .	{ "creation_date": "2025-02-03", "file_name": "(Cambridge Studies in Advanced Mathematics 97) Hugh L. Montgomery, Robert C. Vaughan - Multiplicative number theory I_ Classical theory-Cambridge University Press (2006).pdf", "file_path": "/home/suryaremanan/Documents/IMO/augmentoolkit/original/input/(Cambridge Studies in Advanced Mathematics 97) Hugh L. Montgomery, Robert C. Vaughan - Multiplicative number theory I_ Classical theory-Cambridge University Press (2006).pdf", "file_size": 2365282, "file_type": "application/pdf", "last_modified_date": "2025-02-03", "page_label": "266" }
8.4 Beurling’s generalized prime numbers 267 Since this latter integral is uniformly convergent for σ> 1 − 1/d + δ, we de- duce that ζK (s) is analytic in the half-plane σ> 1 − 1/d apart from a simple pole at s = 1 with residue c. Moreover, we see that if δ is ﬁxed, δ> 0, then ζK (s) ≪\| t \| uniformly for σ ≥ 1 − 1/d + δ, \|t \|≥ 1. If a and b are two ideals in OK , then N (ab) = N (a) N (b). (8.44) Hence ζK (s) has an Euler product formula ζK (s) = ∏ p (1 − N (p)−s )−1 for σ> 1. On taking logarithmic derivatives we also see that − ζ′ K ζK (s) = ∑ a /Lambda1 (a) N (a)−s where /Lambda1 (a) = log N (p)i f a = pk , /Lambda1 (a) = 0 otherwise. Thus, as in Lemma 6.5, ℜ ( −3 ζ′ K ζK (σ) − 4 ζ′ K ζK (σ + it ) − ζ′ K ζK (σ + 2it ) ) ≥ 0 for σ> 1 and any real t . Also as in Chapter 6 we may derive a zero-free region for ζK (s), namely that ζK (s) ̸=0 provided that σ> 1 − c/log τ. Here, as before, τ =\| t \|+ 4, and c is a constant depending on K . Continuing as in Chapter 6, we can derive estimates analogous to those in Theorem 6.7, but with constants depending onK , and we may use our quantitative version of Perron’s formula (Theorem 5.2) to establish a quantitative version of the Prime Ideal Theorem: Theorem 8.9 (Landau) Let K be an algebraic number ﬁeld of ﬁnite de- gree over Q, and let OK denote the ring of algebraic integers in K . Then for x ≥ 2 the number of prime ideals p in OK such that N (p) ≤ xi s li(x ) + OK (x exp(−c√ log x )) where c depends on K . It is notable that the chain of reasoning we have just described depends only on the estimate (8.43) and the identity (8.44). Thus the entire situation could be abstracted as follows. Suppose we have a sequenceP of real numbers pi such that 1 < p1 ≤ p2 ≤· · · and pi →∞ . W e call these numbers ‘generalized primes’. W e form products of powers of these numbers, pa1 1 pa2 2 ··· pak k , and call such products ‘generalized integers’. Let N (x ) denote the number of such products whose value does not exceed x .I f N (x ) = cx + O (x θ) (8.45)	{ "creation_date": "2025-02-03", "file_name": "(Cambridge Studies in Advanced Mathematics 97) Hugh L. Montgomery, Robert C. Vaughan - Multiplicative number theory I_ Classical theory-Cambridge University Press (2006).pdf", "file_path": "/home/suryaremanan/Documents/IMO/augmentoolkit/original/input/(Cambridge Studies in Advanced Mathematics 97) Hugh L. Montgomery, Robert C. Vaughan - Multiplicative number theory I_ Classical theory-Cambridge University Press (2006).pdf", "file_size": 2365282, "file_type": "application/pdf", "last_modified_date": "2025-02-03", "page_label": "267" }
268 Further discussion of the Prime Number Theorem for some c > 0 and θ< 1, then by the reasoning we have outlined it follows that the number P (x ) of generalized primes pi such that pi ≤ x is li( x ) + O (x exp(−c√log x )). The integers Z form an additive group, a cyclic group generated by the number 1. Moreover, the positive integers form a multiplicative semigroup with the primes as generators. From the additive property of the integers we know that [x ] = x + O (1), which is a strong form of (8.45). However, it is now quite clear that our proof of the Prime Number Theorem requires no further knowledge of the additive nature of the integers beyond this estimate. W e have seen that the estimate (8.45) gives a generalization of the Prime Number Theorem with the classical error term. W e now consider the issue of how much this hypothesis can be weakened, if the goal is only to obtain a generalization of (8.1), namely thatP (x ) ∼ x /log x as x →∞ . Theorem 8.10 (Beurling) Let P ={ pi } where 1 < p1 ≤ p2 ≤· · · and p i → ∞, and let N (x ) denote the number of products p a1 1 pa2 2 ··· pak k ≤ x where the ai are non-negative integers. Suppose that there is a positive constant c such that N (x ) = cx + O ( x (log x )γ ) (8.46) for x ≥ 2. Let P (x ) denote the number of members of P not exceeding x . If γ> 3/2, then P (x ) ∼ x log x (8.47) as x →∞ . Proof Let N ={ n j } where 1 = n1 < n2 ≤ n3 ≤· · · are the generalized inte- gers, and for σ> 1 let ζP(s) = ∑ n∈N n−s . Since the n ∈ N are not necessarily rational integers, the above is not necessarily an ordinary Dirichlet series, but it is an example of a ‘generalized Dirichlet series’. In any case it is an absolutely convergent series and by integration by parts as in the proof of Theorem 1.3 we see that ζP(s) = ∫ ∞ 1− u−s dN (u) = s ∫ ∞ 1 N (u)u−s−1 du . W e subtract cu from N (u) to see that ζP(s) = cs s − 1 + s ∫ ∞ 1 ( N (u) − cu )u−s−1 du .	{ "creation_date": "2025-02-03", "file_name": "(Cambridge Studies in Advanced Mathematics 97) Hugh L. Montgomery, Robert C. Vaughan - Multiplicative number theory I_ Classical theory-Cambridge University Press (2006).pdf", "file_path": "/home/suryaremanan/Documents/IMO/augmentoolkit/original/input/(Cambridge Studies in Advanced Mathematics 97) Hugh L. Montgomery, Robert C. Vaughan - Multiplicative number theory I_ Classical theory-Cambridge University Press (2006).pdf", "file_size": 2365282, "file_type": "application/pdf", "last_modified_date": "2025-02-03", "page_label": "268" }
8.4 Beurling’s generalized prime numbers 269 From (8.46) we know that ∫∞ 1 \|N (u) − cu \|u−2 du < ∞. Hence the integral above is uniformly convergent for σ ≥ 1, and consequently it is continuous in this closed half-plane. Thus we can extend the deﬁnition of ζP(s) so that ζP(s) = c/(s − 1) + r0 (s) and r0 (s) is continuous for σ ≥ 1. T o bound the modulus of continuity of r0 (s) we differentiate. Thus ζ′ P(s) =− c/(s − 1)2 + r1 (s) for σ> 1 where r1 (s) = r ′ 0 (s) = ∫ ∞ 1 ( N (u) − cu )u−s−1 du − s ∫ ∞ 1 ( N (u) − cu )(log u)u−s−1 du . If (8.46) holds with γ> 2, then ∫∞ 1 \|N (u) − cu \|(log u)u−2 du < ∞ and then r1 (s) is continuous in the closed half-plane σ ≥ 1. When γ is smaller, however, the situation is more delicate. From now on we assume, as we may , that 3 /2 < γ ≤ 2. Since ∫ ∞ 2 (log u)1−γu−σ du = ∫ ∞ log 2 v1−γe−(σ−1)v d v = (σ − 1)γ−2 ∫ ∞ (σ−1) log 2 u1−γe−u du ≪ (σ − 1)− 1 2 +η, where η = η(γ) > 0, from (8.46) we deduce that r1 (s) ≪ (σ − 1)− 1 2 +η uni- formly for σ> 1. Consequently , if t is ﬁxed, t ̸=0, then ζP(σ + it ) − ζP(1 + it ) = ∫ σ 1 ζ′ P(α + it ) d α ≪ (σ − 1) 1 2 +η (8.48) for σ> 1, σ near 1. Next we use the above estimate to show that ζP(1 + it ) ̸=0 (8.49) when t is real, t ̸=0. By mimicking the proof of the usual Euler product formula for ζ(s), we see that ζP(s) = ∏ p∈P (1 − p−s )−1 for σ> 1. This product is absolutely convergent, and each factor is non-zero, so ζP(s) ̸=0 for σ> 1, and indeed we may write log ζP(s) = ∑ p∈P ∞∑ r =1 1 r p−rs . (8.50) Instead of the cosine polynomial 3 + 4 cos θ + cos 2 θ used in Chapter 6, we must now employ a non-negative cosine polynomial a0 + ∑ K k=1 ak cos kθ for which the ratio a1 /a0 is larger. As we observed in Section 6.1, it is always the	{ "creation_date": "2025-02-03", "file_name": "(Cambridge Studies in Advanced Mathematics 97) Hugh L. Montgomery, Robert C. Vaughan - Multiplicative number theory I_ Classical theory-Cambridge University Press (2006).pdf", "file_path": "/home/suryaremanan/Documents/IMO/augmentoolkit/original/input/(Cambridge Studies in Advanced Mathematics 97) Hugh L. Montgomery, Robert C. Vaughan - Multiplicative number theory I_ Classical theory-Cambridge University Press (2006).pdf", "file_size": 2365282, "file_type": "application/pdf", "last_modified_date": "2025-02-03", "page_label": "269" }
270 Further discussion of the Prime Number Theorem case that a1 < 2a0 , but we can make a1 as close to 2 a0 as we wish by using the Fej´ er kernel/Delta1 K (θ) with K large, since /Delta1 K (θ) = 1 + 2 K∑ k=1 ( 1 − k K ) cos 2 πkθ = 1 K (sin πK θ sin πθ )2 ≥ 0. Hence if σ> 1, then K∏ k=−K ζP(σ + ikt )(1−\|k\|/K ) = exp (∑ p∈P ∞∑ r =1 1 rp r σ K∑ k=−K (1 −\| k\|/K ) p−irkt ) = exp (∑ p∈P ∞∑ r =1 1 rp r σ /Delta1 K (rt (log p)/(2π)) ) . Now ζP(σ − it ) = ζP(σ + it ), so that \|ζP(σ − it )\|=\| ζP(σ + it )\|. Also, /Delta1 K (θ) ≥ 0 for all θ. Hence from the above we see that ζP(σ) K∏ k=1 \|ζP(σ + ikt )\|2(1−k/K ) ≥ 1. Suppose that t is a ﬁxed, non-zero real number. As σ tends to 1 from above, the numbers \|ζP(σ + ikt )\| tend to ﬁnite limits, and ζP(σ) ≍ 1/(σ − 1). Thus \|ζP(σ + it )\|≫ (σ − 1) K 2(K −1) as σ → 1+. Here the implicit constant may depend not only on P but also on t . Suppose now that ζP(1 + it ) = 0. Then from (8.48) we have ζP(σ + it ) ≪ (σ − 1) 1 2 +η as σ → 1+. This contradicts the lower bound above if K is large enough, say K > 1 + 1 2η. Hence ζ(1 + it ) ̸=0, as desired. For n ∈ N let /Lambda1 (n) = log p if n = pr and p ∈ P, /Lambda1 (n) = 0 otherwise. On differentiating (8.50) we see that − ζ′ P ζP (s) = ∑ n∈N /Lambda1 (n)n−s for σ> 1. Set S(x ) = ∑ n∈N n≤x /Lambda1 (n). Suppose for the moment that γ> 2. Then r0 (s) and r1 (s) are both continuous in the closed half-plane σ ≥ 1, and then − ζ′ P ζP (s) = 1 s − 1 + r (s) where r (s) =− r0 (s) + (s − 1)r1 (s) (s − 1)ζP(s)	{ "creation_date": "2025-02-03", "file_name": "(Cambridge Studies in Advanced Mathematics 97) Hugh L. Montgomery, Robert C. Vaughan - Multiplicative number theory I_ Classical theory-Cambridge University Press (2006).pdf", "file_path": "/home/suryaremanan/Documents/IMO/augmentoolkit/original/input/(Cambridge Studies in Advanced Mathematics 97) Hugh L. Montgomery, Robert C. Vaughan - Multiplicative number theory I_ Classical theory-Cambridge University Press (2006).pdf", "file_size": 2365282, "file_type": "application/pdf", "last_modified_date": "2025-02-03", "page_label": "270" }
8.4 Beurling’s generalized prime numbers 271 is continuous in the closed half-plane σ ≥ 1. Then by the Wiener–Ikehara theorem it follows that S(x ) ∼ x as x →∞ . Under the weaker hypothesis that 3/2 <γ ≤ 2 we are no longer able to guarantee that r1 (s) is continuous, but by Plancherel’s identity it is bounded in mean-square. Thus, below , we follow the lines of the proof of the Wiener–Ikehara theorem, but with an appeal to Plancherel’s identity where continuity had sufﬁced before. Suppose that δ> 0, that T is a large positive number, and that E (u) is deﬁned as in Lemma 8.5. Then∑ n∈N n≤x /Lambda1 (n)n−δ = x ∑ n∈N /Lambda1 (n)n−1−δE (log n − log x ) which by Lemma 8.5 is ≤ x ∑ n∈N /Lambda1 (n)n−1−δ f+(log n − log x ) ≤ x ∑ n∈N /Lambda1 (n)n−1−δ ∫ T −T ˆf +(t ) (x n )−2πit dt =− x ∫ T −T ˆf +(t )x −2πit ζ′ P ζP (1 + δ − 2πit ) dt . (8.51) As for the main term, we note that similarly ∫ ∞ 1 u−1−δ f+(log u − log x ) du = ∫ ∞ 1 u−1−δ ∫ T −T ˆf +(t ) (x u )−2πit du dt = ∫ T −T ˆf +(t )x −2πit ∫ ∞ 1 u−1−δ+2πit du dt = ∫ T −T ˆf +(t )x −2πit 1 δ − 2πit dt . W e multiply both sides of this by x and combine with (8.51) to see that ∑ n∈N n≤x /Lambda1 (n)n−δ ≤ x ∫ ∞ 1 u−1−δ f+(log u − log x ) du (8.52) + x ∫ T −T ˆf +(t )x −2πit ( − ζ′ P ζP (1 + δ − 2πit ) − 1 δ − 2πit ) dt . By using our formulæ for ri (s) in terms of integrals we see that we may write r1 (s) = r ′ 0 (s) =− sJ (s) + r0 (s) − c s where J (s) = ∫ ∞ 1 ( N (u) − cu ) (log u)u−s−1 du ,	{ "creation_date": "2025-02-03", "file_name": "(Cambridge Studies in Advanced Mathematics 97) Hugh L. Montgomery, Robert C. Vaughan - Multiplicative number theory I_ Classical theory-Cambridge University Press (2006).pdf", "file_path": "/home/suryaremanan/Documents/IMO/augmentoolkit/original/input/(Cambridge Studies in Advanced Mathematics 97) Hugh L. Montgomery, Robert C. Vaughan - Multiplicative number theory I_ Classical theory-Cambridge University Press (2006).pdf", "file_size": 2365282, "file_type": "application/pdf", "last_modified_date": "2025-02-03", "page_label": "271" }
272 Further discussion of the Prime Number Theorem and −ζ′ P(s) = c (s − 1)2 − r0 (s) − c s + sJ (s). Thus − ζ′ P ζP (s) − 1 s − 1 = c(s − 1) + (1 − 2s)r0 (s) s(s − 1)ζP(s) + s ζP(s) J (s) and by splitting the integral at X , where X is a large parameter we have − ζ′ P ζP (s) − 1 s − 1 = C (s) + R(s) where R(s) = ∫ ∞ X ( N (u) − cu ) (log u)u−s−1 du and C (s) is continuous for σ ≥ 1. W e consider ﬁrst the contribution of the remainder R(s) to (8.52). By the Cauchy–Schwartz inequality we see that ⏐ ⏐ ⏐ ⏐ ∫ T −T ˆf +(t )x −2πit R(1 + δ − 2πit ) dt ⏐ ⏐ ⏐ ⏐ 2 (8.53) ≤ ∫ T −T ⏐ ⏐ ⏐ˆf +(t ) 1 + δ − 2πit ζP(1 + δ − 2πit ) ⏐ ⏐ ⏐ 2 dt ∫ T −T ⏐ ⏐ ⏐ ∫ ∞ X ( N (u) − cu )(log u) u2+δ−2πit du ⏐ ⏐ ⏐ 2 dt . In Theorem 5.4 we take σ = 1 + δ and w(u) = ( N (u) − cu ) log u for u ≥ X , w(u) = 0 otherwise. Thus we see that ∫ ∞ −∞ ⏐ ⏐ ⏐ ⏐ ∫ ∞ X ( N (u) − cu )(log u)u−2−δ+2πit du ⏐ ⏐ ⏐ ⏐ 2 dt = ∫ ∞ X ( N (u) − cu )2 (log u)2 u−3−2δ du , which by (8.46) is ≪ ∫ ∞ X u−1 (log u)2−2γ du ≪γ (log X )3−2γ uniformly for δ> 0. The ﬁrst integral on the right-hand side of (8.53) is also uniformly bounded as δ tends to 0, since ζP(1 + it ) ̸=0. Thus the contribution of R(s) to (8.52) is ≪γ (log X )3/2−γ, uniformly for δ> 0. Hence if we let δ tend to 0 from above in (8.52), and divide through by x , we ﬁnd that S(x ) x ≤ ∫ ∞ 1 u−1 f+(log u − log x ) du + ∫ T −T ˆf +(t )x −2πit C (1 − 2πit ) dt + Oγ ( (log X )3/2−γ) .	{ "creation_date": "2025-02-03", "file_name": "(Cambridge Studies in Advanced Mathematics 97) Hugh L. Montgomery, Robert C. Vaughan - Multiplicative number theory I_ Classical theory-Cambridge University Press (2006).pdf", "file_path": "/home/suryaremanan/Documents/IMO/augmentoolkit/original/input/(Cambridge Studies in Advanced Mathematics 97) Hugh L. Montgomery, Robert C. Vaughan - Multiplicative number theory I_ Classical theory-Cambridge University Press (2006).pdf", "file_size": 2365282, "file_type": "application/pdf", "last_modified_date": "2025-02-03", "page_label": "272" }
8.4 Beurling’s generalized prime numbers 273 As x tends to inﬁnity , the ﬁrst integral on the right tends to ∫∞ −∞ f+(v) d v. Since ˆf +(t )C (1 − 2πit ) is a continuous function of t , by the Riemann–Lebesgue lemma the second integral on the right tends to 0 as x tends to inﬁnity . Hence lim sup x →∞ S(x ) x ≤ ∫ ∞ −∞ f+(v) d v + Oγ ( (log X )3/2−γ) . By Lemma 8.5 we know that the integral on the right is < 1 + ε if T is sufﬁ- ciently large. Since X may also be taken arbitrarily large, we conclude that the limsup above is ≤ 1. By a similar argument with f+ replaced by f−, we ﬁnd that the corresponding liminf is ≥ 1, so we have the generalized Prime Number Theorem in the form S(x ) ∼ x . By integrating by parts we obtain the desired relation (8.47). □ W e now show that the exponent 3 /2 is critical in Beurling’s theorem. Theorem 8.11 The primes P can be chosen in such a way that (8.46) holds with γ = 3/2 but (8.47) fails. The general idea is that if ζP(s) has a simple pole at s = 1 and zeros of multiplicity 1 /2a t1 ± ia , say ζP(s) = (s − 1 − ia )1/2 (s − 1 + ia )1/2 s − 1 H (s) (8.54) where H (s) is analytic for σ>θ , θ< 1, then we can express N (x ) by Perron’s formula applied to ζP(s). After moving the contour to the left, we would ﬁnd that the residue at s = 1 gives rise to the main term cx , and the loop of contour around the branch points at 1 ± ia give oscillatory terms of size x /(log x )3/2 . On the other hand, − ζ′ P ζP (s) = 1 s − 1 − 1 2(s − 1 − ia ) − 1 2(s − 1 + ia ) − H ′ H (s), which suggests that S is approximately x − x 1+ia 2(1 + ia ) − x 1−ia 2(1 − ia ) . This is of the order of magnitude x but not asymptotic to x . It is of course essen- tial that the above main term should be increasing; we note that its derivative is 1− cos(a log x ) ≥ 0. For a rigorous construction we begin by deﬁning primes so that S(x ) approximates this main term, and then we show that the resulting ζP(s) satisﬁes (8.54). Proof Let a be a ﬁxed positive real number, and set f (x ) = ∫ x 1 1 − cos(a log u) log u du .	{ "creation_date": "2025-02-03", "file_name": "(Cambridge Studies in Advanced Mathematics 97) Hugh L. Montgomery, Robert C. Vaughan - Multiplicative number theory I_ Classical theory-Cambridge University Press (2006).pdf", "file_path": "/home/suryaremanan/Documents/IMO/augmentoolkit/original/input/(Cambridge Studies in Advanced Mathematics 97) Hugh L. Montgomery, Robert C. Vaughan - Multiplicative number theory I_ Classical theory-Cambridge University Press (2006).pdf", "file_size": 2365282, "file_type": "application/pdf", "last_modified_date": "2025-02-03", "page_label": "273" }
274 Further discussion of the Prime Number Theorem W e note that this function is increasing and tends to inﬁnity with x . Hence for each positive integer j there is a unique real number p j such that f ( p j ) = j .I f p j ≤ x < p j +1 , then P (x ) = j and j ≤ f (x ) < j + 1; hence P (x ) = [ f (x )]. By integration by parts we see that ∫ x 2 ui α log u du = x 1+i α (1 + i α) log x + O ( x (log x )2 ) . By taking α =− a,0,a, and combining, we see that f (x ) = ( 1 − x ia 2(1 + ia ) − x −ia 2(1 − ia ) ) x log x + O ( x (log x )2 ) , and consequently lim inf x →∞ P (x ) x /log x = 1 − 1√ 1 + a2 , lim sup x →∞ P (x ) x /log x = 1 + 1√ 1 + a2 . Clearly ∑ p∈ P p≤x log p = ∫ x 1 log ud [ f (u)] = ∫ x 1 log udf (u) − ∫ x 1 log ud { f (u)} = ∫ x 1 1 − cos(a log u) du − [ { f (u)} log u ⏐ ⏐ ⏐ x 1 + ∫ x 1 { f (u)} u du = x − x 1+ia 2(1 + ia ) − x 1−ia 2(1 − ia ) + O (log x ), and hence S(x ) = x − x 1+ia 2(1 + ia ) − x 1−ia 2(1 − ia ) + O ( x 1/2 ) . Let r (x ) denote this last error term. Then for σ> 1, − ζ′ P ζP (s) = ∫ ∞ 1 u−s dS (u) = 1 s − 1 − 1 2(s − 1 − ia ) − 1 2(s − 1 + ia ) + g(s) where g(s) is analytic for σ> 1/2. Hence log ζP(s) =− log(s − 1) + 1 2 log(s − 1 − ia ) + 1 2 log(s − 1 + ia ) + G (s) where G ′(s) =− g(s), and so we have (8.54) with H (s) = eG (s) . T o complete the proof we need not only (8.54) but also an estimate of the size of ζP(s) when σ< 1. T o this end we mimic the approach used to estimate	{ "creation_date": "2025-02-03", "file_name": "(Cambridge Studies in Advanced Mathematics 97) Hugh L. Montgomery, Robert C. Vaughan - Multiplicative number theory I_ Classical theory-Cambridge University Press (2006).pdf", "file_path": "/home/suryaremanan/Documents/IMO/augmentoolkit/original/input/(Cambridge Studies in Advanced Mathematics 97) Hugh L. Montgomery, Robert C. Vaughan - Multiplicative number theory I_ Classical theory-Cambridge University Press (2006).pdf", "file_size": 2365282, "file_type": "application/pdf", "last_modified_date": "2025-02-03", "page_label": "274" }
8.4 Beurling’s generalized prime numbers 275 1/ζ(s) in Theorem 6.7. Since P (x ) ≪ x /log x it follows that log ζP(1 + δ + it ) ≪ log 1 /δuniformly for 0 <δ ≤ 1/2. If t ≥ 4 + a and 1 − 1/log t ≤ σ ≤ 1 + 1/log t , then − ζ′ P ζP (s) = ∑ n≤t 2 n∈N /Lambda1 (n)n−s + ∫ ∞ t 2 u−s dS (u). Here the sum is ≪ ∑ n≤t 2 n∈N /Lambda1 (n) n ≪ log t , and the integral is t 2(1−s) s − 1 − t 2(1+ia −s) 2(s − 1 − ia ) − t 2(1−ia −s) 2(s − 1 + ia ) − r (t 2 ) t 2s + s ∫ ∞ t 2 r (u)u−s−1 du ≪ 1, so that log ζP(s) =− ∫ 1+1/log t σ ζ′ P ζP (α + it )d α + log ζP(1 + 1/log t + it ) ≪ 1 + log log t for σ ≥ 1 − 1/log t . Hence there is a constant A such that ζP(s) ≪ (log t ) A for σ ≥ 1 − 1/log t , t ≥ 4 + a. W e now estimate N (x ) by taking an inverse Mellin transform of ζP(s). However, the truncated Perron formula (Corollary 5.3) is not so useful since we lack information concerning the number of generalized integers in a short interval. T o avoid this difﬁculty we use Ces` aro weights as discussed in Section 5.1, by means of which we see that ifb > 1 and h > 0, then 1 2πih ∫ b+i ∞ b−i ∞ ζP(s) (x + h)s+1 − x s+1 s(s + 1) ds = ∑ n∈N w+(n) where w+(u) = ⎧ ⎨ ⎩ 1( u ≤ x ), (x + h − u)/h (x < u ≤ x + h), 0( u > x + h). W e now pull the contour to the left. In view of (8.54), at s = 1 we encounter a simple pole with residue c(x + h/2) where c = aH (1). Because of the branch points at 1 ± ia , we slit the plane by the segments σ ± ia for −∞ <σ ≤ 1. Our contour follows the upper and lower sides of these segments; the integral along these loops is≪ ∫1 −∞(x + h)σ(1 − σ)1/2 d σ ≪ x /(log x )3/2 . By taking	{ "creation_date": "2025-02-03", "file_name": "(Cambridge Studies in Advanced Mathematics 97) Hugh L. Montgomery, Robert C. Vaughan - Multiplicative number theory I_ Classical theory-Cambridge University Press (2006).pdf", "file_path": "/home/suryaremanan/Documents/IMO/augmentoolkit/original/input/(Cambridge Studies in Advanced Mathematics 97) Hugh L. Montgomery, Robert C. Vaughan - Multiplicative number theory I_ Classical theory-Cambridge University Press (2006).pdf", "file_size": 2365282, "file_type": "application/pdf", "last_modified_date": "2025-02-03", "page_label": "275" }
276 Further discussion of the Prime Number Theorem more care, and using Theorem C.3, we could obtain oscillatory main terms of this order of magnitude. On the rest of the contour we estimate the integral as in the proof of the Prime Number Theorem, and thus we see that N (x ) ≤ ∑ n∈N w+(n) = cx + 1 2 ch + O ( x (log x )3/2 ) + O (x 2 h exp ( − C √ log x )) . On taking h = x /(log x )2 we obtain an upper bound of the desired type. T o obtain a corresponding lower bound we argue similarly from the formula 1 2πih ∫ b+i ∞ b−i ∞ ζP(s) x s+1 − (x − h)s+1 s(s + 1) ds = ∑ n∈N w−(n) where w−(u) = ⎧ ⎨ ⎩ 1( u ≤ x − h), (x − u)/h (x − h < u ≤ x ), 0( u ≥ x ). □ 8.5 Notes Section 8.1. Historical accounts of the development of prime number theory and of the various proofs of the Prime Number Theorem have been given by Bateman & Diamond (1996), Narkiewicz (2000), and by Schwarz (1994). Axer’s theorem originates in Axer (1911). The deﬁnitive account of Axer’s theorem is that of Landau (1912). Section 8.2. In former times, an argument was considered to be ‘non- elementary’ if it involved Cauchy’s theorem or Fourier inversion. Prior to Sel- berg’s elementary proof of the Prime Number Theorem, a distinction was drawn between those results that could be obtained by elementary arguments, and those that could not. Selberg’s elementary proof rendered the terminology nugatory . Theorem 8.3 and a deduction of the Prime Number Theorem occur in Selberg (1949). There are a number of variants of the less than straightforward T auberian process used in the deduction; see, for example, Erd ˝ os (1949), Wright (1952), and Levinson (1969). For a historical review of elementary proofs of the Prime Number Theorem see Goldfeld (2004). Quantitative estimates of the form π(x ) = li(x )(1 + O ((log x )−a )) have been derived by elementary methods. van der Corput (1956) obtained a = 1/200, Kuhn (1955) obtained a = 1/10, Breusch a = 1/6 − ε, and	{ "creation_date": "2025-02-03", "file_name": "(Cambridge Studies in Advanced Mathematics 97) Hugh L. Montgomery, Robert C. Vaughan - Multiplicative number theory I_ Classical theory-Cambridge University Press (2006).pdf", "file_path": "/home/suryaremanan/Documents/IMO/augmentoolkit/original/input/(Cambridge Studies in Advanced Mathematics 97) Hugh L. Montgomery, Robert C. Vaughan - Multiplicative number theory I_ Classical theory-Cambridge University Press (2006).pdf", "file_size": 2365282, "file_type": "application/pdf", "last_modified_date": "2025-02-03", "page_label": "276" }
8.5 Notes 277 Wirsing (1962) a = 3/4. Then Bombieri (1962a,b) and Wirsing (1964) showed that the above is true for any ﬁxed positive a. Subsequently , elementary tech- niques have been used to show that π(x ) = li(x ) + O (x exp(−c(log x )−b )) for various values of b. Diamond & Steinig (1970) obtained b = 1/7 − ε, Lavrik & Sobirov (1973) b = 1/6 − ε, and Srinivasan & Sampath (1988) b = 1/6. Although the estimates obtained by elementary methods have thus far been weaker than those derived by analytic means, we have no reason to believe that this will always be the case. Section 8.3. The theorem of Ikehara (1931) represented a major advance, because it gave for the ﬁrst time a T auberian theorem that could be used to prove the Prime Number Theorem without imposing growth conditions on the Dirichlet series generating function. Ikehara assumed thatα(s) − c/(s − 1) is analytic in the closed half-plane σ ≥ 1. Wiener (1932) showed that mere conti- nuity is enough, but this is of lesser signiﬁcance, since still weaker hypotheses are sufﬁcient – see Korevaar (2006). The heart of the Wiener–Ikehara proof of the Prime Number Theorem is Lemma 8.5, which has the effect of enabling one to reduce directly to a use of the Riemann–Lebesgue lemma on a ﬁnite section of the lineℜs = 1. In the proof of Lemma 8.5 we see that it sufﬁces to take T = C /ε, and from Exercise 8.3.5 we see that it is necessary to take T ≥ 1/(2ε) + O (1). Graham & V aaler (1981) have shown that f+ and f− can be constructed so that equality is achieved in Exercise 8.3.5(e),(g). Lemma 8.5, with T small and ε large, is also useful for proving interesting theorems of Fatou and Riesz. Fatou (1906) showed that if an = o(1), then the series f (z) = ∑ an zn converges at any point of the circle \|z\|= 1 at which f is analytic. Landau (1910, Section 10) gives Riesz’s proof that if ∑ n≤x an = o(x ), then the Dirichlet series α(s) = ∑ an n−s converges at every point of the line σ = 1 at which α(s) is analytic. Riesz (1916) extended this to generalized Dirichlet series. For detailed discussion of Wiener’s T auberian theorem, the Ikehara theorem, and T auberian theorems associated with the elementary proof of the Prime Number Theorem see Pitt (1958). Section 8.4. The concept of generalized primes are introduced in Beurling (1937). The hypothesis of Theorem 8.10 can be weakened: Kahane (1997) has shown that if∫ ∞ 1 ( N (x ) − cx )2 x −3 (log x )2 dx < ∞, then (8.47) still follows.	{ "creation_date": "2025-02-03", "file_name": "(Cambridge Studies in Advanced Mathematics 97) Hugh L. Montgomery, Robert C. Vaughan - Multiplicative number theory I_ Classical theory-Cambridge University Press (2006).pdf", "file_path": "/home/suryaremanan/Documents/IMO/augmentoolkit/original/input/(Cambridge Studies in Advanced Mathematics 97) Hugh L. Montgomery, Robert C. Vaughan - Multiplicative number theory I_ Classical theory-Cambridge University Press (2006).pdf", "file_size": 2365282, "file_type": "application/pdf", "last_modified_date": "2025-02-03", "page_label": "277" }
278 Further discussion of the Prime Number Theorem Theorem 8.11 is due to Diamond (1970b). Diamond (1973) also showed that if (8.46) holds with γ> 1, then one has an estimate P (x ) ≪ x /log x of the Chebyshev kind. Zhang (1993) showed that the hypothesis here can be weakened to ∫ ∞ 1 sup y≤x \|N ( y) − cy \| y dx x < ∞ . In the negative direction, Hall (1973) showed that if γ< 1, then the hypothesis (8.46) is not sufﬁcient to imply a Chebyshev estimate. Also, Kahane (1998) has shown that the hypothesis ∫ ∞ 1 \|N (x ) − cx \| x 2 dx < ∞ does not imply a Chebyshev estimate. Zhang (1987b) has shown that if (8.46) holds withγ> 1, then ∑ n≤x n∈N µ(n) = o(x ) . In the classical context, the above is equivalent – by Axer’s theorem – to the Prime Number Theorem. However, in the Beurling situation, if 1<γ ≤ 3/2, the above holds but PNT may fail. Nyman (1949) showed that if (8.46) holds for all γ (with the implicit con- stant depending on γ), then P (x ) = li(x ) + Oc (x /(log x )c ) for all c. Malliavin (1961) showed that if N (x ) = cx + O (x exp(−(log x )a )) where 0 < a < 1, then π(x ) = li(x ) + O (x exp(−(log x )b )) with b = a/10. Both these authors proved converse theorems in which an estimate for P (x ) is used to estab- lish a corresponding estimate for N (x ), but those results have since been sharpened by Diamond (1970a). It is now known that the method of Lan- dau, in which one starts from (8.45) to derive the indicated error term, is sharp: Diamond, Montgomery & V orhauer (2006) have shown that ifθ is given, 1/2 <θ< 1, then there exists a Beurling system for which (8.45) holds, but P (x ) − li(x ) = /Omega1 ±(x exp(−c√ log x )). Some of the ideas and themes developed in connection with the Prime Num- ber Theorem have had ramiﬁcations in surprisingly diverse areas. See, for exam- ple, Hejhal’s expositions (1976, 1983) of Selberg’s trace formula forPSL (2,R), and the monograph of Parry & Pollicott (1990) on the periodic orbit structure of hyperbolic dynamics. Some writers avoid the term ‘Beurling’, and instead discuss ‘arithmetic semigroups’. The mathematics is the same in either case. For more on this topic see Bateman & Diamond (1969), and Knopfmacher (1990).	{ "creation_date": "2025-02-03", "file_name": "(Cambridge Studies in Advanced Mathematics 97) Hugh L. Montgomery, Robert C. Vaughan - Multiplicative number theory I_ Classical theory-Cambridge University Press (2006).pdf", "file_path": "/home/suryaremanan/Documents/IMO/augmentoolkit/original/input/(Cambridge Studies in Advanced Mathematics 97) Hugh L. Montgomery, Robert C. Vaughan - Multiplicative number theory I_ Classical theory-Cambridge University Press (2006).pdf", "file_size": 2365282, "file_type": "application/pdf", "last_modified_date": "2025-02-03", "page_label": "278" }
8.6 References 279 8.6 References Axer, A. (1911). ¨Uber einige Grenzwerts¨ atze, Sitz. Kais. Akad. Wiss. Wien. math-natur . Klasse 120, 1253–1298. Balanzario, E. P . (2000). On Chebyshev’s inequalities for Beurling’s generalized primes, Math. Slovaca 50, No.4, 415–436. Bateman, P . T . (1972). The distribution of values of the Euler function, Acta Arith. 21, 329–345. Bateman, P . T . & Diamond, H. G. (1969). Asymptotic distribution of Beurling’s generalized prime numbers, Studies in Number Theory , W . J. LeV eque, Ed., MAA Studies in math. 6. W ashington: Mathematical Association of America, pp. 152–210. (1996). A hundred years of prime numbers, Amer . Math. Monthly 103, 729–741. Beurling, A. (1937). Analyse de la loi asymptotique de la distribution des nombres premiers g´ en´ eralis´ es, I,Acta Math. 68, 255–291. Bombieri, E. (1962a). Maggiorazione del resto nel “Primzahlsatz” col metodo di Erd ˝ os– Selberg, Ist. Lombardo Accad. Sci. Lett. Rend. A 96, 343–350. (1962b). Sulle formule di A. Selberg generalizzate per classi di funzioni aritmetiche e le applicazioni al problema del resto nel “Primzahlsatz”, Riv . Mat. Univ . P arma (2) 3, 393–440. Borel, J.-P . (1980/81). Quelques r´ esultats d’´ equir´ epartition li´ es aux nombres g´ en´ eralis´ es de Beurling, Acta Arith . 38, 255–272. (1984). Sur le prolongement des fonctions ζ associ´ ees ` a un syst` eme des nombres premiers g´ en´ eralis´ es de Beurling,Acta Arith . 43, 273–282. Breusch, R. (1960). An elementary proof of the prime number theorem with remainder term, P aciﬁc J. Math . 10, 487–497. van der Corput, J. G. (1956). Sur le reste dans la d´ emonstration ´ el´ ementaire du theor` eme des nombres premiers, Colloque sur la Th ´eorie des Nombres (Bruxelles, 1955). Paris: Masson & Cie, pp. 163–182. Diamond, H. G. (1969). The prime number theorem for Beurling’s generalized numbers, J. Number Theory 1, 200–207. (1970a). Asymptotic distribution of Beurling’s generalized integers, Illinois J. Math . 14, 12–28. (1970b). A set of generalized numbers showing Beurling’s theorem to be sharp, Illinois J. Math . 14, 29–34. (1973). Chebyshev estimates for Beurling generalized prime numbers, Proc. Amer . Math. Soc. 39, 503–508. (1977). When do Beurling generalized integers have a density?, J. Reine Angew . Math . 295, 22–39. Diamond, H. G., Montgomery , H. L., & V orhauer, U. M. A. (2006). Beurling primes with large oscillation, Math. Ann ., 334, 1–36. Diamond, H. G. & Steinig, J. (1970). An elementary proof of the prime number theorem with a remainder term, Invent. Math . 11, 199–258. Dressler, R. E. (1970). A density which counts multiplicity , P aciﬁc Math. J. 34, 371–378. Erd ˝ os, P . (1949). On a new method in elementary number theory which leads to an elementary proof of the prime number theorem, Proc. Natl. Acad. Sci. USA 35, 374–384.	{ "creation_date": "2025-02-03", "file_name": "(Cambridge Studies in Advanced Mathematics 97) Hugh L. Montgomery, Robert C. Vaughan - Multiplicative number theory I_ Classical theory-Cambridge University Press (2006).pdf", "file_path": "/home/suryaremanan/Documents/IMO/augmentoolkit/original/input/(Cambridge Studies in Advanced Mathematics 97) Hugh L. Montgomery, Robert C. Vaughan - Multiplicative number theory I_ Classical theory-Cambridge University Press (2006).pdf", "file_size": 2365282, "file_type": "application/pdf", "last_modified_date": "2025-02-03", "page_label": "279" }
280 Further discussion of the Prime Number Theorem Fatou, P . (1906). S´ eries trigonom´ etriques et s´ eries de T aylor, Acta Math . 30, 335–400. Goldfeld, D. (2004). The elementary proof of the prime number theorem: an histori- cal perspective, Number Theory (New Y ork, 2003). New Y ork: Springer-V erlag, pp. 179–192. Graham, S. W . & V aaler, J. D. (1981). A class of extremal functions for the Fourier transform, Trans. Amer . Math. Soc . 265, 283–302. Hall, R. S. (1972). The prime number theorem for generalized primes, J. Number Theory 4, 313–320. (1973). Beurling generalized prime number systems in which the Chebyshev inequal- ities fail, Proc. Amer . Math. Soc . 40, 79–82. Hejhal, D. A. (1976). The Selberg Trace F ormula for P S L (2,R). V ol. I, Lecture Notes Math. 548. Berlin: Springer-V erlag. (1983). The Selberg Trace F ormula for P S L (2,R). V ol. 2, Lecture Notes Math. 1001. Berlin: Springer-V erlag. Ikehara, S. (1931). An extension of Landau’s theorem in the analytic theory of numbers, J. Math. Phys . 10, 1–12. Ingham, A. E. (1945). Some T auberian theorems connected with the prime number theorem, J. London Math. Soc . 20, 171–180. Kahane, J.-P . (1995). Sur travaux de Beurling et Malliavin, S´eminaire Bourbaki V ol. 7 Exp. 225, Paris: Soc. Math. France, 27–39. (1996). Une formula de Fourier pour les nombres premiers. Application aux nombres premiers g´ en´ eralis´ es de Beurling,Harmonic analysis from the Pichorides viewpoint (Anogia, 1995) Publ. Math. Orsay, 96–01, Orsay: Univ . Paris XI, 41–49. (1997). Sur les nombres premiers g´ en´ eralis´ es de Beurling. Preuve d’une conjecture de Bateman et Diamond, J. Th ´eor . Nombres Bordeaux 9, 251–266. (1998). Le r ˆ ole des alg` ebres A de Wiener, A∞ de Beurling et H 1 de Sobolev dans la th´ eorie des nombres premiers g´ en´ eralis´ es de Beurling, Ann. Inst. F ourier (Grenoble) 48, 611–648. (1999). Un th´ eor` eme de Littlewood pour les nombres premiers de Beurling Bull. London Math. Soc . 31, 424–430. Knopfmacher, J. (1990). Abstract Analytic Number Theory , Second Edition. New Y ork: Dover. Korevaar, J. (2006). The Wiener–Ikehara theorem by complex analysis, Proc. Amer . Math. Soc. 134, 1107–1116. Kuhn, P . (1955). Eine V erbesserung des Restgliedes beim elementaren Beweis des Primzahlsatzes, Math. Scand . 3, 75–89. Landau, E. (1910). ¨Uber die Bedeutung einiger neuen Grenswerts¨ atze der Herren Hardy und Axer, Prace mat.-ﬁz. 21, 97–177; Collected W orks, V ol. 4. Essen: Thales V erlag, 1986, pp. 267–347. (1912). ¨Uber einige neuere Grenzwerts¨ atze, Rend. Circ. Mat. P alermo 34, 121–131; Collected W orks, V ol. 5. Essen: Thales V erlag, 1986, pp. 145–155. Lavrik, A. F . & Sobirov , A. ˇS. (1973). The remainder term in the elementary proof of the Prime Number Theorem, Dokl. Akad. Nauk SSSR 211, 534–536. Levinson, N. (1969). A motivated account of an elementary proof of the Prime Number Theorem, Amer . Math. Monthly 76, 225–245. Malliavin, P . (1961). Sur le reste de la loi asymptotique de r´ epartition des nombres premiers g´ en´ eralis´ es de Beurling,Acta Math . 106, 281–298.	{ "creation_date": "2025-02-03", "file_name": "(Cambridge Studies in Advanced Mathematics 97) Hugh L. Montgomery, Robert C. Vaughan - Multiplicative number theory I_ Classical theory-Cambridge University Press (2006).pdf", "file_path": "/home/suryaremanan/Documents/IMO/augmentoolkit/original/input/(Cambridge Studies in Advanced Mathematics 97) Hugh L. Montgomery, Robert C. Vaughan - Multiplicative number theory I_ Classical theory-Cambridge University Press (2006).pdf", "file_size": 2365282, "file_type": "application/pdf", "last_modified_date": "2025-02-03", "page_label": "280" }
8.6 References 281 Narkiewicz, W . (2000). The Development of Prime Number Theory . Berlin: Springer- V erlag. Nyman, B. (1949). A general Prime Number Theorem, Acta Math . 81, 299–307. Parry , W . & Pollicott, M. (1990). Zeta functions and the periodic orbit structure of hyperbolic dynamics, Ast´erisque No. 268 , pp. 187–188. Pitt, H. R. (1958). T auberian Theorems. Oxford: Oxford University Press. Riesz, M. (1916). Ein Konvergenzsatz f ¨ ur Dirichletsche Reihen, Acta Math . 40, 349–361. Schwarz, W . (1994). Some remarks on the history of the Prime Number Theorem from 1896 to 1960, Development of mathematics 1900–1950 (Luxembourg, 1992). Basel: Birkh¨ auser, pp. 565–616. Selberg, A. (1949). An elementary proof of the prime-number theorem, Ann. Math . (2) 50, 305–313. Srinivasan, B. R. & Sampath, A. (1988). An elementary proof of the Prime Number Theorem with a remainder term, J. Indian, Math. Soc ., New Ser. 53, No.1-4, 1-50. Widder, D. V . (1971). An Introduction to Transform Theory . New Y ork: Academic Press. Wiener, N. (1932). T auberian theorems, Ann. of Math. (2) 33, 1–100; Collected W orks, V ol. 2. Cambridge: MIT , 1979, pp. 519–619. Wirsing, E. (1962). Elementare Beweise des Primzahlsatzes mit Restglied, I, J. Reine Angew . Math. 211, 205–214. (1964). Elementare Beweise des Primzahlsatzes mit Restglied, II, Reine Angew ., J. Math. 214/215, 1–18. Wright, E. M. (1952). The elementary proof of the Prime Number Theorem, Proc. Roy. Soc. Edinbugh A 63, 257–267. Zhang, W . B. (1987a). Chebyshev type estimates for Beurling generalized prime num- bers, Proc. Amer . Math. Soc . 101, 205–212. (1987b). A generalization of Hal´ asz’s theorem to Beurling’s generalized integers and its application, Illinois J. Math . 31, 645–664. (1988). Density and O -density of Beurling generalized integers, J. Number Theory 30, 120–139. (1993). Chebyshev type estimates for Beurling generalized prime numbers, II, Trans. Amer . Math. Soc. 337, 651–675.	{ "creation_date": "2025-02-03", "file_name": "(Cambridge Studies in Advanced Mathematics 97) Hugh L. Montgomery, Robert C. Vaughan - Multiplicative number theory I_ Classical theory-Cambridge University Press (2006).pdf", "file_path": "/home/suryaremanan/Documents/IMO/augmentoolkit/original/input/(Cambridge Studies in Advanced Mathematics 97) Hugh L. Montgomery, Robert C. Vaughan - Multiplicative number theory I_ Classical theory-Cambridge University Press (2006).pdf", "file_size": 2365282, "file_type": "application/pdf", "last_modified_date": "2025-02-03", "page_label": "281" }

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