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CAMBRIDGE STUDIES IN ADVANCED MATHEMATICS 97
Editorial Board
B. Bollobas, W . Fulton, A. Katok, F . Kirwan, P . Sarnak, B. Simon, B. T otaro
MUL TIPLICA TIVE NUMBER THEOR Y I:
CLASSICAL THEOR Y
Prime numbers are the multiplicative building blocks of natural numbers. Un-
derstanding their overall influence and especially their distribution gives rise
to central questions in mathematics and physics. In particular their finer distri-
bution is closely connected with the Riemann hypothesis, the most important
unsolved problem in the mathematical world. Assuming only subjects covered
in a standard degree in mathematics, the authors comprehensively cover all the
topics met in first courses on multiplicative number theory and the distribution
of prime numbers. They bring their extensive and distinguished research exper-
tise to bear in preparing the student for intelligent reading of the more advanced
research literature. The text, which is based on courses taught successfully over
many years at Michigan, Imperial College and Pennsylvania State, is enriched
by comprehensive historical notes and references as well as over 500 exercises.
Hugh Montgomery is a Professor of Mathematics at the University of Michigan.
Robert V aughan is a Professor of Mathematics at Pennsylvannia State
University . | {
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CAMBRIDGE STUDIES IN ADV ANCED MA THEMA TICS
All the titles listed below can be obtained from good booksellers of from Cambridge University
Press. For a complete series listing visit:
http://www .cambridge.org/series/sSeries.asp?code=CSAM
Already published
70 R. Iorio & V . Iorio F ourier analysis and partial differential equations
71 R. Blei Analysis in integer and fractional dimensions
72 F . Borceaux & G. Janelidze Galois theories
73 B. Bollob´ as Random graphs
74 R. M. Dudley Real analysis and probability
75 T . Sheil-Small Complex polynomials
76 C. V oisin Hodge theory and complex algebraic geometry, I
77 C. V oisin Hodge theory and complex algebraic geometry, II
78 V . Paulsen Completely bounded maps and operator algebras
79 F . Gesztesy & H. Holden Soliton Equations and Their Algebro-Geometric Solution, I
81 S. Mukai An Introduction to Invariants and Moduli
82 G. T ourlakis Lectures in Logic and Set Theory, I
83 G. T ourlakis Lectures in Logic and Set Theory, II
84 R. A. Bailey Association Schemes
85 J. Carlson, S. M ¨ uller-Stach & C. Peters P eriod Mappings and P eriod Domains
86 J. J. Duistermaat & J. A. C. Kolk Multidimensional Real Analysis I
87 J. J. Duistermaat & J. A. C. Kolk Multidimensional Real Analysis II
89 M. Golumbic & A. Trenk T olerance Graphs
90 L. Harper Global Methods for Combinatorial Isoperimetric Problems
91 I. Moerdijk & J. Mrcun Introduction to F oliations and Lie Groupoids
92 J. Kollar, K. E. Smith & A. Corti Rational and Nearly Rational V arieties
93 D. Applebaum Levy Processes and Stochastic Calculus
94 B. Conrad Modular F orms and the Ramanujan Conjecture
95 M. Schechter An Introduction to Nonlinear Analysis
96 R. Carter Lie Algebras of Finite and Affine T ype
97 H. L. Montgomery & R. C V aughan Multiplicative Number Theory I
98 I. Chavel Riemannian Geometry
99 D. Goldfeld Automorphic F orms and L-Functions for the Group GL(n,R)
100 M. Marcus & J. Rosen Markov Processes, Gaussian Processes, and Local Times
101 P . Gille & T . Szamuely Central Simple Algebras and Galois Cohomology
102 J. Bertoin Random Fragmentation and Coagulation Processes | {
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Multiplicative Number Theory
I. Classical Theory
HUGH L. MONTGOMERY
University of Michigan, Ann Arbor
ROBERT C. VAUGHAN
P ennsylvania State University, University P ark | {
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cambridge university press
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Cambridge University Press
The Edinburgh Building, Cambridge cb2 2ru, UK
First published in print format
isbn-13 978-0-521-84903-6
isbn-13 978-0-511-25746-9
© Cambridge University Press 2006
2006
Information on this title: www.cambrid ge.org/9780521849036
This publication is in copyright. Subject to statutory exception and to the provision of
relevant collective licensing agreements, no reproduction of any part may take place
without the written permission of Cambridge University Press.
isbn-10 0-511-25746-5
isbn-10 0-521-84903-9
Cambridge University Press has no responsibility for the persistence or accuracy of urls
for external or third-party internet websites referred to in this publication, and does not
guarantee that any content on such websites is, or will remain, accurate or appropriate.
Published in the United States of America by Cambridge University Press, New York
www.cambridge.org
hardback
eBook (NetLibrary)
eBook (NetLibrary)
hardback | {
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Dedicated to our teachers :
P . T . Bateman
J. H. H. Chalk
H. Davenport
T . Estermann
H. Halberstam
A. E. Ingham | {
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T alet ¨ar t ¨ankandets b ¨orjan och slut.
Med tanken f ¨oddes talet.
Ut ¨ofver talet n˚ ar tanken icke.
Numbers are the beginning and end of thinking.
With thoughts were numbers born.
Beyond numbers thought does not reach.
Magnus Gustaf Mittag-Leffler, 1903 | {
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Contents
Preface page xi
List of notation xiii
1 Dirichlet series: I 1
1.1 Generating functions and asymptotics 1
1.2 Analytic properties of Dirichlet series 11
1.3 Euler products and the zeta function 19
1.4 Notes 31
1.5 References 33
2 The elementary theory of arithmetic functions 35
2.1 Mean values 35
2.2 The prime number estimates of Chebyshev and of Mertens 46
2.3 Applications to arithmetic functions 54
2.4 The distribution of /Omega1 (n) − ω(n) 65
2.5 Notes 68
2.6 References 71
3 Principles and first examples of sieve methods 76
3.1 Initiation 76
3.2 The Selberg lambda-squared method 82
3.3 Sifting an arithmetic progression 89
3.4 T win primes 91
3.5 Notes 101
3.6 References 104
4 Primes in arithmetic progressions: I 108
4.1 Additive characters 108
4.2 Dirichlet characters 115
4.3 Dirichlet L -functions 120
vii | {
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viii Contents
4.4 Notes 133
4.5 References 134
5 Dirichlet series: II 137
5.1 The inverse Mellin transform 137
5.2 Summability 147
5.3 Notes 162
5.4 References 164
6 The Prime Number Theorem 168
6.1 A zero-free region 168
6.2 The Prime Number Theorem 179
6.3 Notes 192
6.4 References 195
7 Applications of the Prime Number Theorem 199
7.1 Numbers composed of small primes 199
7.2 Numbers composed of large primes 215
7.3 Primes in short intervals 220
7.4 Numbers composed of a prescribed number of primes 228
7.5 Notes 239
7.6 References 241
8 Further discussion of the Prime Number Theorem 244
8.1 Relations equivalent to the Prime Number Theorem 244
8.2 An elementary proof of the Prime Number Theorem 250
8.3 The Wiener–Ikehara T auberian theorem 259
8.4 Beurling’s generalized prime numbers 266
8.5 Notes 276
8.6 References 279
9 Primitive characters and Gauss sums 282
9.1 Primitive characters 282
9.2 Gauss sums 286
9.3 Quadratic characters 295
9.4 Incomplete character sums 306
9.5 Notes 321
9.6 References 323
10 Analytic properties of the zeta function and L -functions 326
10.1 Functional equations and analytic continuation 326
10.2 Products and sums over zeros 345
10.3 Notes 356
10.4 References 356 | {
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Contents ix
11 Primes in arithmetic progressions: II 358
11.1 A zero-free region 358
11.2 Exceptional zeros 367
11.3 The Prime Number Theorem for arithmetic
progressions 377
11.4 Applications 386
11.5 Notes 391
11.6 References 393
12 Explicit formulæ 397
12.1 Classical formulæ 397
12.2 W eil’s explicit formula 410
12.3 Notes 416
12.4 References 417
13 Conditional estimates 419
13.1 Estimates for primes 419
13.2 Estimates for the zeta function 433
13.3 Notes 447
13.4 References 449
14 Zeros 452
14.1 General distribution of the zeros 452
14.2 Zeros on the critical line 456
14.3 Notes 460
14.4 References 461
15 Oscillations of error terms 463
15.1 Applications of Landau’s theorem 463
15.2 The error term in the Prime Number Theorem 475
15.3 Notes 482
15.4 References 484
APPENDICES
A The Riemann–Stieltjes integral
486
A.1 Notes 492
A.2 References 493
B Bernoulli numbers and the Euler–MacLaurin
summation formula 495
B.1 Notes 513
B.2 References 517 | {
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x Contents
C The gamma function 520
C.1 Notes 531
C.2 References 533
D T opics in harmonic analysis 535
D.1 Pointwise convergence of Fourier series 535
D.2 The Poisson summation formula 538
D.3 Notes 542
D.4 References 542
Name index 544
Subject index 550 | {
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Preface
Our object is to introduce the interested student to the techniques, results, and
terminology of multiplicative number theory . It is not intended that our discus-
sion will always reach the research frontier. Rather, it is hoped that the material
here will prepare the student for intelligent reading of the more advanced re-
search literature.
Analytic number theorists are not very uniformly distributed around the
world and it possible that a student may be working without the guidance of an
experienced mentor in the area. With this in mind, we have tried to make this
volume as self-contained as possible.
W e assume that the reader has some acquaintance with the fundamentals of
elementary number theory , abstract algebra, measure theory , complex analysis,
and classical harmonic analysis. More specialized or advanced background
material in analysis is provided in the appendices.
The relationship of exercises to the material developed in a given section
varies widely . Some exercises are designed to illustrate the theory directly
whilst others are intended to give some idea of the ways in which the theory can
be extended, or developed, or paralleled in other areas. The reader is cautioned
that papers cited in exercises do not necessarily contain a solution.
This volume is the first instalment of a larger project. W e are preparing a
second volume, which will cover such topics as uniform distribution, bounds for
exponential sums, a wider zero-free region for the Riemann zeta function, mean
and large values of Dirichlet polynomials, approximate functional equations,
moments of the zeta function andL functions on the line σ = 1/2, the large
sieve, V inogradov’s method of prime number sums, zero density estimates,
primes in arithmetic progressions on average, sums of primes, sieve methods,
the distribution of additive functions and mean values of multiplicative func-
tions, and the least prime in an arithmetic progression. The present volume was
xi | {
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xii Preface
twenty-five years in preparation—we hope to be a little quicker with the second
volume.
Many people have assisted us in this work—including P . T . Bateman, E.
Bombieri, T . Chan, J. B. Conrey , H. G. Diamond, T . Estermann, J. B. Friedlan-
der, S. W . Graham, S. M. Gonek, A. Granville, D. R. Heath-Brown, H. Iwaniec,
H. Maier, G. G. Martin, D. W . Masser, A. M. Odlyzko, G. Peng, C. Pomerance,
H.–E. Richert, K. Soundararajan, and U. M. A. V orhauer. In particular, our
doctoral students, and their students also, have been most helpful in detecting
errors of all types. W e are grateful to them all. W e would be most happy to hear
from any reader who detects a misprint, or might suggest improvements.
Finally we thank our loved ones and friends for their long term support
and the long–suffering David Tranah at Cambridge University Press for his
forbearance. | {
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Notation
Symbol Meaning Found on page
CThe set of complex numbers. 109
F p A field of p elements. 9
N The set of natural numbers, 1, 2, ... 114
Q The set of rational numbers. 120
R The set of real numbers. 43
TR /Z, known as the circle group or
the one-dimensional torus , which is
to say the real numbers modulo 1.
110
Z The set of rational integers. 20
B constant in the Hadamard product
forξ(s)
347, 349
Bk Bernoulli numbers. 496ff
Bk (x ) Bernoulli polynomials. 45, 495ff
B (χ) constant in the Hadamard product
for ξ(s,χ )
351, 352
C0 Euler’s constant 26
cq (n) The sum of e(an /q ) with a running
over a reduced residue system
moduloq ; known as Ramanujan’s
sum.
110
cχ(n) = ∑ q
a=1 χ(a)e(an /q ). 286, 290
d (n) The number of positive divisors of n,
called the divisor function .
2
dk (n) The number of ordered k-tuples of
positive integers whose product
isn.
43
E0 (χ) = 1i f χ = χ0 , 0 otherwise. 358
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xiv List of notation
Symbol Meaning Found on page
Ek The Euler numbers , also known as
the secant coefficients .
506
e(θ) = e2πi θ; the complex exponential
with period 1.
64, 108ff
L (s,χ ) A Dirichlet L -function. 120
Li(x ) =
∫x
0
du
log u with the Cauchy
principal value taken at 1; the
logarithmic integral.
189
li(x ) =
∫x
2
du
log u ; the logarithmic
integral.
5
M (x ) = ∑
n≤x µ(n) 182
M (x ; q ,a) The sum of µ(n) over those n ≤ x
for which n ≡ a (mod q ).
383
M (x ,χ ) The sum of χ(n)µ(n) over those
n ≤ x .
383
N (T ) The number of zeros ρ = β + i γ
of ζ(s) with 0 <γ ≤ T.
348, 452ff
N (T ,χ ) The number of zeros ρ = β + i γ
of L (s,χ ) with β> 0 and
0 ≤ β ≤ T.
454
P (n) The largest prime factor of n. 202
Q(x ) the number of square-free numbers
not exceeding x
36
S(t ) = 1
π argζ( 1
2 + it ). 452
S(t ,χ ) = 1
π argL ( 1
2 + it ,χ ). 454
si(x ) =−
∫∞
x
sin u
u du ; the sine integral . 139
Tk The tangent coefficients . 505
w(u) The Buchstab function , defined by
the equation ( uw(u))′ = w(u − 1)
for u > 2 together with the initial
condition w(u) = 1/u for
1 < u ≤ 2.
216
Z (t ) Hardy’s function. The function
Z (t ) is real-valued, and
|Z (t )|=| ζ( 1
2 + it )|.
456ff
β The real part of a zero of the zeta
function or of anL -function.
173
Ŵ(s) =
∫∞
0 e−x x s−1 dx for σ> 0;
called the Gamma function .
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List of notation xv
Symbol Meaning Found on page
Ŵ(s,a) =
∫∞
a e−wws−1 d w; the incomplete
Gamma function .
327
γ The imaginary part of a zero of the
zeta function or of anL -function.
172
/Delta1 N (θ) = 1 + 2 ∑ N −1
n=1 (1 − n/N ) cos 2 πnθ;
known as the Fe j´er kernel .
174
ε(χ) = τ(χ)/
(
i κq 1/2 )
. 332
ζ(s) = ∑ ∞
n=1 n−s for σ> 1, known as the
Riemann zeta function .
2
ζ(s,α) = ∑ ∞
n=0 (n + α)−s for σ> 1; known
as the Hurwitz zeta function .
30
ζK (s) ∑
a N (a)−s ; known as the Dedekind
zeta function of the algebraic number
field K .
343
/Theta1 = sup ℜ ρ 430, 463
ϑ(x ) = ∑
p≤x log p.4 6
ϑ(z) = ∑ ∞
n=−∞ e−πn2 z for ℜz > 0. 329
ϑ(x ; q ,a) The sum of log p over primes p ≤ x
for which p ≡ a (mod q ).
128, 377ff
ϑ(x ,χ ) = ∑
p≤x χ( p) log p. 377ff
κ = (1 − χ(−1))/2. 332
/Lambda1 (n) = log p if n = pk , = 0 otherwise;
known as the von Mangoldt Lambda
function.
23
/Lambda1 2 (n) = /Lambda1 (n) log n + ∑
bc=n /Lambda1 (b)/Lambda1 (c). 251
/Lambda1 (x ; q ,a) The sum of λ(n) over those n ≤ x
such that n ≡ a (mod q ).
383
/Lambda1 (x ,χ ) = ∑
n≤x χ(n)λ(n). 383
λ(n) = (−1)/Omega1 (n) ; known as the Liouville
lambda function .
21
µ(n) = (−1)ω(n) for square-free n, = 0
otherwise. Known as the M ¨obius mu
function.
21
µ(σ) the Lindel ¨ of mu function 330
ξ(s) = 1
2 s(s − 1)ζ(s)Ŵ(s/2)π−s/2 . 328
ξ(s,χ ) = L (s,χ )Ŵ((s + κ)/2)(q /π)(s+κ)/2
where χ is a primitive character
modulo q , q > 1.
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xvi List of notation
Symbol Meaning Found on page
/Pi1(x ) = ∑
n≤x /Lambda1 (n)/log n. 416
π(x ) The number of primes not exceeding x .3
π(x ; q ,a) The number of p ≤ x such that p ≡ a
(mod q ),.
90, 358
π(x ,χ ) = ∑
p≤x χ( p). 377ff
ρ = β + i γ; a zero of the zeta function or
of an L -function.
173
ρ(u) The Dickman function , defined by the
equation uρ′(u) =− ρ(u − 1) for u > 1
together with the initial condition
ρ(u) = 1 for 0 ≤ u ≤ 1.
200
σ(n) The sum of the positive divisors of n.2 7
σa (n) = ∑
d |n d a .2 8
τ =| t |+ 4. 14
τ(χ) = ∑ q
a=1 χ(a)e(a/q ); known as the
Gauss sum of χ.
286ff
/Phi1 q (z) The q th cyclotomic polynomial, which is
to say a monic polynomial with integral
coefficients, of degreeϕ(q ), whose roots
are the numbers e(a/q ) for ( a,q ) = 1.
64
/Phi1 (x ,y) The number of n ≤ x such that all prime
factors of n are ≥ y.
215
/Phi1 ( y) = 1
√
2π
∫y
−∞ e−t 2 /2 dt ; the cumulative
distribution function of a normal random
variable with mean 0 and variance 1.
235
ϕ(n) The number of a,1 ≤ a ≤ n, for which
(a,n) = 1; known as Euler’s totient
function.
27
χ(n) A Dirichlet character. 115
ψ(x ) = ∑
n≤x /Lambda1 (n). 46
ψ(x ,y) The number of n ≤ x composed entirely
of primes p ≤ y.
199
ψ(x ; q ,a) The sum of /Lambda1 (n) over n ≤ x for which
n ≡ a (mod q ).
128, 377ff
ψ(x ,χ ) = ∑
n≤x χ(n)/Lambda1 (n). 377ff
/Omega1 (n) The number of prime factors of n,
counting multiplicity .
21
ω(n) The number of distinct primes dividing n.2 1 | {
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List of notation xvii
Symbol Meaning Found on page
[x ] The unique integer such that
[x ] ≤ x < [x ] + 1; called the integer
part of x .
15, 24
{x }= x − [x ]; called the fractional part of x .2 4
∥ x ∥ The distance from x to the nearest
integer.
477
f (x ) = O (g(x )) | f (x )|≤ Cg (x ) where C is an absolute
constant.
3
f (x ) = o(g(x )) lim f (x )/g(x ) = 0. 3
f (x ) ≪ g(x ) f (x ) = O (g(x )). 3
f (x ) ≫ g(x ) g(x ) = O ( f (x )), g non-negative. 4
f (x ) ≍ g(x ) cf (x ) ≤ g(x ) ≤ Cf (x ) for some positive
absolute constants c, C .
4
f (x ) ∼ g(x ) lim f (x )/g(x ) = 1. 3 | {
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|
1
Dirichlet series: I
1.1 Generating functions and asymptotics
The general rationale of analytic number theory is to derive statistical informa-
tion about a sequence{an } from the analytic behaviour of an appropriate gen-
erating function, such as a power series ∑ an zn or a Dirichlet series ∑ an n−s .
The type of generating function employed depends on the problem being in-
vestigated. There are no rigid rules governing the kind of generating function
that is appropriate – the success of a method justifies its use – but we usually
deal with additive questions by means of power series or trigonometric sums,
and with multiplicative questions by Dirichlet series. For example, if
f (z) =
∞∑
n=1
znk
for |z| < 1, then the nth power series coefficient of f (z)s is the number rk,s (n)
of representations of n as a sum of s positive kth powers,
n = mk
1+ mk
2+···+ mk
s.
W e can recover rk,s (n) from f (z)s by means of Cauchy’s coefficient formula:
rk,s (n) = 1
2πi
∮ f (z)s
zn+1 dz .
By choosing an appropriate contour, and estimating the integrand, we can de-
termine the asymptotic size ofrk,s (n)a s n →∞ , provided that s is sufficiently
large, say s > s0 (k). This is the germ of the Hardy–Littlewood circle method,
but considerable effort is required to construct the required estimates.
T o appreciate why power series are useful in dealing with additive prob-
lems, note that if A(z) = ∑ ak zk and B (z) = ∑ bm zm then the power series
1 | {
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2 Dirichlet series: I
coefficients of C (z) = A(z) B (z) are given by the formula
cn =
∑
k+m=n
ak bm . (1.1)
The terms are grouped according to the sum of the indices, because
zk zm = zk+m .
A Dirichlet series is a series of the form α(s) = ∑ ∞
n=1 an n−s where s is
a complex variable. If β(s) = ∑ ∞
m=1 bm m−s is a second Dirichlet series and
γ(s) = α(s)β(s), then (ignoring questions relating to the rearrangement of terms
of infinite series)
γ(s) =
∞∑
k=1
ak k−s
∞∑
m=1
bm m−s =
∞∑
k=1
∞∑
m=1
ak bm (km )−s =
∞∑
n=1
(∑
km =n
ak bm
)
n−s .
(1.2)
That is, we expect that γ(s) is a Dirichlet series, γ(s) = ∑ ∞
n=1 cn n−s , whose
coefficients are
cn =
∑
km =n
ak bm . (1.3)
This corresponds to (1.1), but the terms are now grouped according to the
product of the indices, sincek−s m−s = (km )−s .
Since we shall employ the complex variable s extensively , it is useful to have
names for its real and complex parts. In this regard we follow the rather peculiar
notation that has become traditional:s = σ + it .
Among the Dirichlet series we shall consider is the Riemann zeta function ,
which for σ> 1 is defined by the absolutely convergent series
ζ(s) =
∞∑
n=1
n−s . (1.4)
As a first application of (1.3), we note that if α(s) = β(s) = ζ(s) then the
manipulations in (1.3) are justified by absolute convergence, and hence we see
that
∞∑
n=1
d (n)n−s = ζ(s)2 (1.5)
for σ> 1. Here d (n)i st h e divisor function , d (n) = ∑
d |n 1.
From the rate of growth or analytic behaviour of generating functions we
glean information concerning the sequence of coefficients. In expressing our
findings we employ a special system of notation. For example, we say , ‘f (x )i s
asymptotic to g(x )’ as x tends to some limiting value (say x →∞ ), and write | {
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1.1 Generating functions and asymptotics 3
f (x ) ∼ g(x )( x →∞ ), if
lim
x →∞
f (x )
g(x ) = 1.
An instance of this arises in the formulation of the Prime Number Theorem
(PNT), which concerns the asymptotic size of the numberπ(x ) of prime num-
bers not exceeding x ; π(x ) = ∑
p≤x 1. Conjectured by Legendre in 1798, and
finally proved in 1896 independently by Hadamard and de la V all´ ee Poussin,
the Prime Number Theorem asserts that
π(x ) ∼ x
log x .
Alternatively , we could say that
π(x ) = (1 + o(1)) x
log x ,
which is to say that π(x )i s x /log x plus an error term that is in the limit
negligible compared with x /log x . More generally , we say , ‘ f (x ) is small oh
of g(x )’, and write f (x ) = o(g(x )), if f (x )/g(x ) → 0a s x tends to its limit.
The Prime Number Theorem can be put in a quantitative form,
π(x ) = x
log x + O
( x
(log x )2
)
. (1.6)
Here the last term denotes an implicitly defined function (the difference be-
tween the other members of the equation); the assertion is that this function has
absolute value not exceedingCx (log x )−2 . That is, the above is equivalent to
asserting that there is a constant C > 0 such that the inequality
⏐
⏐
⏐π(x ) − x
log x
⏐
⏐
⏐≤ Cx
(log x )2
holds for all x ≥ 2. In general, we say that f (x ) is ‘big oh of g(x )’, and write
f (x ) = O (g(x )) if there is a constant C > 0 such that | f (x )|≤ Cg (x ) for all
x in the appropriate domain. The function f may be complex-valued, but g
is necessarily non-negative. The constant C is called the implicit constant ;
it is an absolute constant unless the contrary is indicated. For example, if C
is liable to depend on a parameter α, we might say , ‘For any fixed value of
α, f (x ) = O (g(x ))’. Alternatively , we might say , ‘ f (x ) = O (g(x )) where the
implicit constant may depend on α’, or more briefly , f (x ) = Oα(g(x )).
When there is no main term, instead of writing f (x ) = O (g(x )) we save a
pair of parentheses by writing instead f (x ) ≪ g(x ). This is read, ‘ f (x ) is less-
than-less-than g(x )’, and we write f (x ) ≪α g(x ) if the implicit constant may
depend on α. T o provide an example of this notation, we recall that Chebyshev | {
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4 Dirichlet series: I
0
0000
0000
0000
0000
00000 00000 00000 00000 1000000
Figure 1.1 Graph of π(x ) (solid) and x /log x (dotted) for 2 ≤ x ≤ 106 .
proved that π(x ) ≪ x /log x . This is of course weaker than the Prime Number
Theorem, but it was derived much earlier, in 1852. Chebyshev also showed
thatπ(x ) ≫ x /log x . In general, we say that f (x ) ≫ g(x ) if there is a positive
constant c such that f (x ) ≥ cg(x ) and g is non-negative. In this situation both
f and g take only positive values. If both f ≪ g and f ≫ g then we say that f
and g have the same order of magnitude, and write f ≍ g. Thus Chebyshev’s
estimates can be expressed as a single relation,
π(x ) ≍ x
log x .
The estimate (1.6) is best possible to the extent that the error term is not
o(x (log x )−2 ). W e have also a special notation to express this:
π(x ) − x
log x = /Omega1
( x
(log x )2
)
.
In general, if lim sup x →∞ | f (x )|/g(x ) > 0 then we say that f (x ) is ‘Omega of
g(x )’, and write f (x ) = /Omega1 (g(x )). This is precisely the negation of the statement
‘ f (x ) = o(g(x ))’. When studying numerical values, as in Figure 1.1, we find
that the fit of x /log x to π(x ) is not very compelling. This is because the error
term in the approximation is only one logarithm smaller than the main term.
This error term is not oscillatory – rather there is a second main term of this | {
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1.1 Generating functions and asymptotics 5
size:
π(x ) = x
log x + x
(log x )2 + O
( x
(log x )3
)
.
This is also best possible, but the main term can be made still more elaborate to
give a smaller error term. Gauss was the first to propose a better approximation to
π(x ). Numerical studies led him to observe that the density of prime numbers in
the neighbourhood of x is approximately 1 /log x . This suggests that the number
of primes not exceeding x might be approximately equal to the logarithmic
integral,
li(x ) =
∫ x
2
1
log u du .
(Orally , ‘li’ rhymes with ‘pi’.) By repeated integration by parts we can show
that
li(x ) = x
K −1∑
k=1
(k − 1)!
(log x )k + OK
( x
(log x )K
)
for any positive integer K ; thus the secondary main terms of the approximation
to π(x ) are contained in li(x).
In Chapter 6 we shall prove the Prime Number Theorem in the sharper
quantitative form
π(x ) = li(x ) + O
( x
exp(c√log x )
)
for some suitable positive constant c. Note that exp( c√log x ) tends to infinity
faster than any power of log x . The error term above seems to fall far from
what seems to be the truth. Numerical evidence, such as that in T able 1.1,
suggests that the error term in the Prime Number Theorem is closer to√
x in
size. Gauss noted the good fit, and also that π(x ) < li(x ) for all x in the range of
his extensive computations. He proposed that this might continue indefinitely ,
but the numerical evidence is misleading, for in 1914 Littlewood showed that
π(x ) − li(x ) = /Omega1 ±
(x 1/2 log log log x
log x
)
.
Here the subscript ± indicates that the error term achieves the stated or-
der of magnitude infinitely often, and in both signs. In particular, the dif-
ferenceπ − li has infinitely many sign changes. More generally , we write
f (x ) = /Omega1 +(g(x )) if lim sup x →∞ f (x )/g(x ) > 0, we write f (x ) = /Omega1 −(g(x ))
if lim inf x →∞ f (x )/g(x ) < 0, and we write f (x ) = /Omega1 ±(g(x )) if both these re-
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6 Dirichlet series: I
T able 1.1 V alues of π(x ), li(x),x /log x for x = 10k , 1 ≤ k ≤ 22.
x π(x ) li( x ) x /log x
10 4 5.12 4.34
102 25 29.08 21.71
103 168 176.56 144.76
104 1229 1245.09 1085.74
105 9592 9628.76 8685.89
106 78498 78626.50 72382.41
107 664579 664917.36 620420.69
108 5761455 5762208.33 5428681.02
109 50847534 50849233.90 48254942.43
1010 455052511 455055613.54 434294481.90
1011 4118054813 4118066399.58 3948131653.67
1012 37607912018 37607950279.76 36191206825.27
1013 346065536839 346065458090.05 334072678387.12
1014 3204941750802 3204942065690.91 3102103442166.08
1015 29844570422669 29844571475286.54 28952965460216.79
1016 279238341033925 279238344248555.75 271434051189532.39
1017 2623557157654233 2623557165610820.07 2554673422960304.87
1018 24739954287740860 24739954309690413.98 24127471216847323.76
1019 234057667276344607 234057667376222382.22 228576043106974646.13
1020 2220819602560918840 2220819602783663483.55 2171472409516259138.26
1021 21127269486018731928 21127269486616126182.33 20680689614440563221.48
1022 201467286689315906290 201467286691248261498.15 197406582683296285295.97
In the exercises below we give several examples of the use of generating
functions, mostly power series, to establish relations between various counting
functions.
1.1.1 Exercises
1. Let r (n) be the number of ways that n cents of postage can be made, using
only 1 cent, 2 cent, and 3 cent stamps. That is, r (n) is the number of ordered
triples ( x1 ,x2 ,x3 ) of non-negative integers such that x1 + 2x2 + 3x3 = n.
(a) Show that
∞∑
n=0
r (n)zn = 1
(1 − z)(1 − z2 )(1 − z3 )
for |z| < 1.
(b) Determine the partial fraction expansion of the rational function above. | {
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1.1 Generating functions and asymptotics 7
That is, find constants a, b, ... , f so that the above is
a
(z − 1)3 + b
(z − 1)2 + c
z − 1 + d
z + 1 + e
z − ω + f
z − ω
where ω = e2πi /3 and ω = e−2πi /3 are the primitive cube roots of unity .
(c) Show that r (n) is the integer nearest ( n + 3)2 /12.
(d) Show that r (n) is the number of ways of writing n = y1 + y2 + y3 with
y1 ≥ y2 ≥ y3 ≥ 0.
2. Explain why
∞∏
k=0
(
1 + z2k
)
= 1 + z + z2 +···
for |z| < 1.
3. (L. Mirsky & D. J. Newman) Suppose that 0 ≤ ak < mk for 1 ≤ k ≤ K , and
that m1 < m2 < ··· < m K . This is called a family of covering congruences
if every integer x satisfies at least one of the congruences x ≡ ak (mod mk ).
A system of covering congruences is called exact if for every value of x
there is exactly one value of k such that x ≡ ak (mod mk ). Show that if the
system is exact then
K∑
k=1
zak
1 − zmk
= 1
1 − z
for |z| < 1. Show that the left-hand side above is
∼ e2πia K /m K
m K (1 − r )
when z = re 2πi /m K and r → 1−. On the other hand, the right-hand side is
bounded for z in a neighbourhood of e2πi /m K if m K > 1. Deduce that a family
of covering congruences is not exact if mk > 1.
4. Let p(n; k) denote the number of partitions of n into at most k parts, that is, the
number of ordered k-tuples ( x1 ,x2 ,..., xk ) of non-negative integers such
that n = x1 + x2 +···+ xk and x1 ≥ x2 ≥· · ·≥ xk . Let p(n) = p(n; n) de-
note the total number of partitions of n. Also let po (n) be the number of
partitions of n into an odd number of parts, po (n) = ∑
2∤k p(n; k). Finally ,
let pd (n) denote the number of partitions of n into distinct parts, so that
x1 > x2 > ··· > xk . By convention, put p(0) = po (0) = pd (0) = 1.
(a) Show that there are precisely p(n; k) partitions of n into parts not
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8 Dirichlet series: I
(b) Show that
∞∑
n=0
p(n; k)zn =
k∏
j =1
(1 − z j )−1
for |z| < 1.
(c) Show that
∞∑
n=0
p(n)zn =
∞∏
k=1
(1 − zk )−1
for |z| < 1.
(d) Show that
∞∑
n=0
pd (n)zn =
∞∏
k=1
(1 + zk )
for |z| < 1.
(e) Show that
∞∑
n=0
po (n)zn =
∞∏
k=1
(1 − z2k−1 )−1
for |z| < 1.
(f) By using the result of Exercise 2, or otherwise, show that the last two
generating functions above are identically equal. Deduce that po (n) =
pd (n) for all n.
5. Let A(n) denote the number of ways of associating a product of n terms;
thus A(1) = A(2) = 1 and A(3) = 2. By convention, A(0) = 0.
(a) By considering the possible positionings of the outermost parentheses,
show that
A(n) =
n−1∑
k=1
A(k) A(n − k)
for all n ≥ 2.
(b) Let P (z) = ∑ ∞
n=0 A(n)zn . Show that
P (z)2 = P (z) − z.
Deduce that
P (z) = 1 − √
1 − 4z
2 =
∞∑
n=1
(1/2
n
)
22n−1 (−1)n−1 zn .
(c) Conclude that A(n) =
(2n−2
n−1
)
/n for all n ≥ 1. These are called the Cata-
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1.1 Generating functions and asymptotics 9
(d) What needs to be said concerning the convergence of the series used
above?
6. (a) Let nk denote the total number of monic polynomials of degree k in
F p [x ]. Show that nk = pk .
(b) Let P1 ,P2 ,... be the irreducible monic polynomials in F p [x ], listed in
some (arbitrary) order. Show that
∞∏
r =1
(1 + zdeg Pr + z2d e g Pr + z3d e g Pr +··· ) = 1 + pz + p2 z2
+ p3 z3 +···
for |z| < 1/p.
(c) Let gk denote the number of irreducible monic polynomials of degree k
in F p [x ]. Show that
∞∏
k=1
(1 − zk )−gk = (1 − pz )−1 (|z| < 1/p).
(d) T ake logarithmic derivatives to show that
∞∑
k=1
kgk
zk−1
1 − zk = p
1 − pz (|z| < 1/p).
(e) Show that
∞∑
k=1
kgk
∞∑
m=1
zmk =
∞∑
n=1
pn zn (|z| < 1/p).
(f) Deduce that
∑
k|n
kgk = pn
for all positive integers n.
(g) (Gauss) Use the M ¨ obius inversion formula to show that
gn = 1
n
∑
k|n
µ(k) pn/k
for all positive integers n.
(h) Use (f) (not (g)) to show that
pn
n − 2 pn/2
n ≤ gn ≤ pn
n .
(i) If a monic polynomial of degree n is chosen at random from F p [x ], about
how likely is it that it is irreducible? (Assume that p and/or n is large.) | {
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10 Dirichlet series: I
(j) Show that gn > 0 for all p and all n ≥ 1. (If P ∈ F p [x ] is irreducible and
has degree n, then the quotient ring F p [x ]/( P ) is a field of pn elements.
Thus we have proved that there is such a field, for each prime p and
integer n ≥ 1. It may be further shown that the order of a finite field
is necessarily a prime power, and that any two finite fields of the same
order are isomorphic. Hence the field of orderpn , whose existence we
have proved, is essentially unique.)
7. (E. Berlekamp) Let p be a prime number. W e recall that polynomials in a
single variable (mod p) factor uniquely into irreducible polynomials. Thus
a monic polynomial f (x ) can be expressed uniquely (mod p) in the form
g(x )h(x )2 where g(x ) is square-free (mod p) and both g and h are monic. Let
sn denote the number of monic square-free polynomials (mod p) of degree
n. Show that
(∞∑
k=0
sk zk
)(∞∑
m=0
pm z2m
)
=
∞∑
n=0
pn zn
for |z| < 1/p. Deduce that
∞∑
k=0
sk zk = 1 − pz 2
1 − pz ,
and hence that s0 = 1, s1 = p, and that sk = pk (1 − 1/p) for all k ≥ 2.
8. (cf W agon 1987) (a) Let I = [a,b] be an interval. Show that
∫
I e2πix dx = 0
if and only if the length b − a of I is an integer.
(b) Let R = [a,b] × [c,d ] be a rectangle. Show that
∫∫
R e2πi (x +y) dx dy =
0 if and only if at least one of the edge lengths of R is an integer.
(c) Let R be a rectangle that is a union of finitely many rectangles Ri ; the
Ri are disjoint apart from their boundaries. Show that if all the Ri have
the property that at least one of their side lengths is an integer, then R
also has this property .
9. (L. Moser) If A is a set of non-negative integers, let rA(n) denote the number
of representations of n as a sum of two distinct members of A. That is, rA(n)i s
the number of ordered pairs ( a1 ,a2 ) for which a1 ∈ A, a2 ∈ A, a1 + a2 = n,
and a1 ̸=a2 . Let A(z) = ∑
a∈A za .
(a) Show that ∑
n rA(n)zn = A(z)2 − A(z2 ) for |z| < 1.
(b) Suppose that the non-negative integers are partitioned into two sets A
and B in such a way that rA(n) = rB(n) for all non-negative integers n.
Without loss of generality , 0 ∈ A. Show that 1 ∈ B, that 2 ∈ B, and
that 3 ∈ A.
(c) With A and B as above, show that A(z) + B (z) = 1/(1 − z) for |z| < 1.
(d) Show that A(z) − B (z) = (1 − z)
(
A(z2 ) − B (z2 )
)
, and hence by | {
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1.2 Analytic properties of Dirichlet series 11
induction that
A(z) − B (z) =
∞∏
k=0
(
1 − z2k
)
for |z| < 1.
(e) Let the binary weight of n, denoted w(n), be the number of 1’s in the
binary expansion of n. That is, if n = 2k1 +···+ 2kr with k1 > ··· > kr ,
then w(n) = r . Show that A consists of those non-negative integers n
for which w(n) is even, and that B is the set of those integers for which
w(n) is odd.
1.2 Analytic properties of Dirichlet series
Having provided some motivation for the use of Dirichlet series, we now turn to
the task of establishing some of their basic analytic properties, corresponding
to well-known facts concerning power series.
Theorem 1.1Suppose that the Dirichlet series α(s) = ∑ ∞
n=1 an n−s converges
at the point s = s0 , and that H > 0 is an arbitrary constant. Then the series
α(s) is uniformly convergent in the sector S ={ s : σ ≥ σ0 ,|t − t0 |≤ H (σ −
σ0 )}.
By taking H large, we see that the series α(s) converges for all s in the
half-plane σ>σ 0 , and hence that the domain of convergence is a half-plane.
More precisely , we have
Corollary 1.2Any Dirichlet series α(s) = ∑ ∞
n=1 an n−s has an abscissa of
convergence σc with the property that α(s) converges for all s with σ>σ c , and
for no s with σ<σ c . Moreover , if s 0 is a point with σ0 >σ c , then there is a
neighbourhood of s 0 in which α(s) converges uniformly.
In extreme cases a Dirichlet series may converge throughout the plane ( σc =
−∞), or nowhere ( σc =+ ∞ ). When the abscissa of convergence is finite, the
series may converge everywhere on the line σc + it , it may converge at some
but not all points on this line, or nowhere on the line.
Proof of Theorem 1.1Let R(u) = ∑
n>u an n−s0 be the remainder term of the
series α(s0 ). First we show that for any s,
N∑
n=M +1
an n−s = R( M ) M s0 −s − R( N ) N s0 −s + (s0 − s)
∫ N
M
R(u)us0 −s−1 du .
(1.7) | {
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12 Dirichlet series: I
T o see this we note that an = ( R(n − 1) − R(n)) ns0 , so that by partial
summation
N∑
n=M +1
an n−s =
N∑
n=M +1
( R(n − 1) − R(n))ns0 −s
= R( M ) M s0 −s−R( N ) N s0 −s −
N∑
n=M +1
R(n −1)((n −1)s0 −s − ns0 −s ).
The second factor in this last sum can be expressed as an integral,
(n − 1)s0 −s − ns0 −s =− (s0 − s)
∫ n
n−1
us0 −s−1 du ,
and hence the sum is
(s − s0 )
N∑
n=M +1
R(n − 1)
∫ n
n−1
us0 −s−1 du = (s − s0 )
N∑
n=M +1
∫ n
n−1
R(u)us0 −s−1 du
since R(u) is constant in the interval [ n − 1,n). The integrals combine to give
(1.7).
If |R(u)|≤ ε for all u ≥ M and if σ>σ 0 , then from (1.7) we see that
⏐
⏐
⏐
⏐
N∑
n=M +1
an n−s
⏐
⏐
⏐
⏐≤ 2ε+ ε|s − s0 |
∫ ∞
M
uσ0 −σ−1 du ≤
(
2 + |s − s0 |
σ − σ0
)
ε.
For s in the prescribed region we see that
|s − s0 |≤ σ − σ0 +| t − t0 |≤ ( H + 1)(σ − σ0 ),
so that the sum ∑ N
M +1 an n−s is uniformly small, and the result follows by the
uniform version of Cauchy’s principle. □
In deriving (1.7) we used partial summation, although it would have been
more efficient to use the properties of the Riemann–Stieltjes integral (see
Appendix A):
N∑
n=M +1
an n−s =−
∫ N
M
us0 −s dR (u) =− us0 −s R(u)
⏐
⏐
⏐
⏐
N
M
+
∫ N
M
R(u) du s0 −s
by Theorems A.1 and A.2. By Theorem A.3 this is
= M s0 −s R( M ) − N s0 −s R( N ) + (s0 − s)
∫ N
M
R(u)us0 −s−1 du .
In more complicated situations it is an advantage to use the Riemann–Stieltjes
integral, and subsequently we shall do so without apology .
The series α(s) = ∑ an n−s is locally uniformly convergent for σ>σ c , and
each term is an analytic function, so it follows from a general principle of | {
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1.2 Analytic properties of Dirichlet series 13
W eierstrass that α(s) is analytic for σ>σ c , and that the differentiated series is
locally uniformly convergent to α′(s):
α′(s) =−
∞∑
n=1
an (log n)n−s (1.8)
for s in the half-plane σ>σ c .
Suppose that s0 is a point on the line of convergence (i.e., σ0 = σc ), and that
the series α(s0 ) converges. It can be shown by example that
lims→s0
σ>σc
α(s)
need not exist. However, α(s) is continuous in the sector S of Theorem 1.1, in
view of the uniform convergence there. That is,
lims→s0
s∈S
α(s) = α(s0 ), (1.9)
which is analogous to Abel’s theorem for power series.
W e now express a convergent Dirichlet series as an absolutely convergent
integral.
Theorem 1.3Let A (x ) = ∑
n≤x an .I f σc < 0, then A (x ) is a bounded func-
tion, and
∞∑
n=1
an n−s = s
∫ ∞
1
A(x )x −s−1 dx (1.10)
for σ> 0.I f σc ≥ 0, then
lim sup
x →∞
log | A(x )|
log x = σc , (1.11)
and (1.10) holds for σ>σ c .
Proof W e note that
N∑
n=1
an n−s =
∫ N
1−
x −s dA (x ) = A(x )x −s
⏐
⏐
⏐
⏐
N
1−
−
∫ N
1−
A(x ) dx −s
= A( N ) N −s + s
∫ N
1
A(x )x −s−1 dx .
Let φ denote the left-hand side of (1.11). If θ>φ then A(x ) ≪ x θ where the
implicit constant may depend on the an and on θ. Thus if σ>θ , then the integral
in (1.10) is absolutely convergent. Thus we obtain (1.10) by letting N →∞ ,
since the first term above tends to 0 as N →∞ .
Suppose that σc < 0. By Corollary 1.2 we know that A(x ) tends to a finite
limit as x →∞ , and hence φ ≤ 0, so that (1.10) holds for all σ> 0. | {
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14 Dirichlet series: I
Now suppose that σc ≥ 0. By Corollary 1.2 we know that the series in (1.10)
diverges when σ<σ c . Hence φ ≥ σc . T o complete the proof it suffices to show
that φ ≤ σc . Choose σ0 >σ c . By (1.7) with s = 0 and M = 0 we see that
A( N ) =− R( N ) N σ0 + σ0
∫ N
0
R(u)uσ0 −1 du .
Since R(u) is a bounded function, it follows that A( N ) ≪ N σ0 where the implicit
constant may depend on the an and on σ0 . Hence φ ≤ σ0 . Since this holds for
any σ0 >σ c , we conclude that φ ≤ σc . □
The terms of a power series are majorized by a geometric progression at
points strictly inside the circle of convergence. Consequently power series con-
verge very rapidly . In contrast, Dirichlet series are not so well behaved. For
example, the series
∞∑
n=1
(−1)n−1 n−s (1.12)
converges for σ> 0, but it is absolutely convergent only for σ> 1. In general
we let σa denote the infimum of those σ for which ∑ ∞
n=1 |an |n−σ < ∞. Then σa ,
the abscissa of absolute convergence , is the abscissa of convergence of the series∑ ∞
n=1 |an |n−s , and we see that ∑ an n−s is absolutely convergent if σ>σ a ,
but not if σ<σ a . W e now show that the strip σc ≤ σ ≤ σa of conditional
convergence is never wider than in the example (1.12).
Theorem 1.4In the above notation, σc ≤ σa ≤ σc + 1.
Proof The first inequality is obvious. T o prove the second, suppose that ε> 0.
Since the series ∑ an n−σc −ε is convergent, the summands tend to 0, and hence
an ≪ nσc +ε where the implicit constant may depend on the an and on ε. Hence
the series ∑ an n−σc −1−2ε is absolutely convergent by comparison with the series∑ n−1−ε. □
Clearly a Dirichlet series α(s) is uniformly bounded in the half-plane
σ>σ a + ε, but this is not generally the case in the strip of conditional conver-
gence. Nevertheless, we can limit the rate of growth of α(s) in this strip.
T o aid in formulating our next result we introduce a notational convention
that arises because many estimates relating to Dirichlet series are expressed
in terms of the size of|t |. Our interest is in large values of this quantity , but
in order that the statements be valid for small |t | we sometimes write |t |+ 4.
Since this is cumbersome in complicated expressions, we introduce a shorthand:
τ=| t |+ 4. | {
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1.2 Analytic properties of Dirichlet series 15
Theorem 1.5 Suppose that α(s) = ∑ an n−s has abscissa of convergence σc .
If δ and ε are fixed, 0 <ε<δ< 1, then
α(s) ≪ τ1−δ+ε
uniformly for σ ≥ σc + δ. The implicit constant may depend on the coefficients
an ,o n δ, and on ε.
By the example found in Exercise 8 at the end of this section, we see that
the bound above is reasonably sharp.
ProofLet s be a complex number with σ ≥ σc + δ. By (1.7) with s0 = σc + ε
and N →∞ , we see that
α(s) =
M∑
n=1
an n−s + R( M ) M σc +ε−s + (σc + ε− s)
∫ ∞
M
R(u)uσc +ε−s−1 du .
Since the series α(σc + ε) converges, we know that an ≪ nσc +ε, and also that
R(u) ≪ 1. Thus the above is
≪
M∑
n=1
n−δ+ε + M −δ+ε + |σc + ε− s|
σ − σc − ε M σc +ε−σ.
By the integral test the sum here is
<
∫ M
0
u−δ+ε du = M 1−δ+ε
1 − δ + ε ≪ M 1−δ+ε.
Hence on taking M = [τ] we obtain the stated estimate. □
W e know that the power series expansion of a function is unique; we now
show that the same is true for Dirichlet series expansions.
Theorem 1.6If ∑ an n−s = ∑ bn n−s for all s with σ>σ 0 then a n = bn for
all positive integers n.
ProofW e put cn = an − bn , and consider ∑ cn n−s . Suppose that cn = 0 for
all n < N . Since ∑ cn n−σ = 0 for σ>σ 0 we may write
cN =−
∑
n>N
cn ( N /n)σ.
By Theorem 1.4 this sum is absolutely convergent for σ>σ 0 + 1. Since each
term tends to 0 as σ →∞ , we see that the right-hand side tends to 0, by
the principle of dominated convergence. Hence cN = 0, and by induction we
deduce that this holds for all N . □ | {
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16 Dirichlet series: I
Suppose that f is analytic in a domain D, and that 0 ∈ D. Then f can
be expressed as a power series ∑ ∞
n=0 an zn in the disc |z| <r where r is the
distance from 0 to the boundary ∂D of D. Although Dirichlet series are analytic
functions, the situation regarding Dirichlet series expansions is very different:
The collection of functions that may be expressed as a Dirichlet series in some
half-plane is a very special class. Moreover, the lineσc + it of convergence
need not contain a singular point of α(s). For example, the Dirichlet series
(1.12) has abscissa of convergence σc = 0, but it represents the entire function
(1 − 21−s )ζ(s). (The connection of (1.12) to the zeta function is easy to establish,
since
∞∑
n=1
(−1)n−1 n−s =
∞∑
n=1
n−s − 2
∞∑
n=1
n even
n−s = ζ(s) − 21−s ζ(s)
for σ> 1. That this is an entire function follows from Theorem 10.2.) Since a
Dirichlet series does not in general have a singularity on its line of convergence,
it is noteworthy that a Dirichlet series with non-negative coefficients not only
has a singularity on the lineσc + it , but actually at the point σc .
Theorem 1.7 (Landau) Let α(s) = ∑ an n−s be a Dirichlet series whose ab-
scissa of convergence σc is finite. If a n ≥ 0 for all n then the point σc is a
singularity of the function α(s).
It is enough to assume that an ≥ 0 for all sufficiently large n, since any finite
sum ∑ N
n=1 an n−s is an entire function.
Proof By replacing an by an n−σc , we may assume that σc = 0. Suppose that
α(s) is analytic at s = 0, so that α(s) is analytic in the domain D ={ s : σ>
0}∪{ | s| <δ } if δ> 0 is sufficiently small. W e expand α(s) as a power series
at s = 1:
α(s) =
∞∑
k=0
ck (s − 1)k . (1.13)
The coefficients ck can be calculated by means of (1.8),
ck = α(k) (1)
k! = 1
k!
∞∑
n=1
an (− log n)k n−1 .
The radius of convergence of the power series (1.13) is the distance from 1 to
the nearest singularity ofα(s). Since α(s) is analytic in D, and since the nearest
points not in D are ±i δ, we deduce that the radius of convergence is at least√
1 + δ2 = 1 + δ′, say . That is,
α(s) =
∞∑
k=0
(1 − s)k
k!
∞∑
n=1
an (log n)k n−1 | {
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1.2 Analytic properties of Dirichlet series 17
for |s − 1| < 1 + δ′.I f s < 1 then all terms above are non-negative. Since
series of non-negative numbers may be arbitrarily rearranged, for −δ′ < s < 1
we may interchange the summations over k and n to see that
α(s) =
∞∑
n=1
an n−1
∞∑
k=0
(1 − s)k (log n)k
k!
=
∞∑
n=1
an n−1 exp
(
(1 − s) log n
)
=
∞∑
n=1
an n−s .
Hence this last series converges at s =− δ′/2, contrary to the assumption that
σc = 0. Thus α(s) is not analytic at s = 0. □
1.2.1 Exercises
1. Suppose that α(s) is a Dirichlet series, and that the series α(s0 ) is boundedly
oscillating. Show that σc = σ0 .
2. Suppose that α(s) = ∑ ∞
n=1 an n−s is a Dirichlet series with abscissa of con-
vergence σc . Suppose that α(0) converges, and put R(x ) = ∑
n>x an . Show
that σc is the infimum of those numbers θ such that R(x ) ≪ x θ.
3. Let Ak (x ) = ∑
n≤x an (log n)k .
(a) Show that
A0 (x ) − A1 (x )
log x = a1 +
∫ x
2
A1 (u)
u(log u)2 du .
(b) Suppose that A1 (x ) ≪ x θ where θ> 0 and the implicit constant may
depend on the sequence {an }. Show that
A0 (x ) = A1 (x )
log x + O (x θ(log x )−2 ).
(c) Let σc denote the abscissa of convergence of ∑ an n−s , and σ′
c the ab-
scissa of convergence of ∑ an (log n)n−s . Show that σ′
c = σc . (The re-
marks following the proof of Theorem 1.1 imply only that σ′
c ≤ σc .)
4. (Landau 1909b) Let α(s) = ∑ an n−s be a Dirichlet series with abscissa of
convergence σc and abscissa of absolute convergence σa >σ c . Let C (x ) =∑
n≤x an n−σc and A(x ) = ∑
n≤x |an |n−σc .
(a) By a suitable application of Theorem 1.3, or otherwise, show that
C (x ) ≪ x ε and that A(x ) ≪ x σa −σc +ε for any ε> 0, where the implicit
constants may depend on ε and on the sequence {an }.
(b) Show that if σ>σ c then
∑
n>N
an n−s =− C ( N ) N σc −s + (s − σc )
∫ ∞
N
C (u)uσc −s−1 du . | {
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18 Dirichlet series: I
Deduce that the above is ≪ τN σc −σ+ε uniformly for s in the half-plane
σ ≥ σc + ε where the implicit constant may depend on ε and on the
sequence {an }.
(c) Show that
N∑
n=1
|an |n−σ = A( N ) N −σ+σc + (σ − σc )
∫ N
1
A(u)u−σ+σc −1 du
for any σ. Deduce that the above is ≪ N σa −σ+ε uniformly for σ in the
interval σc ≤ σ ≤ σa , for any given ε> 0. Here the implicit constant
may depend on ε and on the sequence {an }.
(d) Let θ(σ) = (σa − σ)/(σa − σc ). By making a suitable choice of N , show
that
α(s) ≪ τθ(σ)+ε
uniformly for s in the strip σc + ε ≤ σ ≤ σa .
5. (a) Show that if α(s) = ∑ an n−s has abscissa of convergence σc < ∞, then
lim
σ→∞
α(σ) = a1 .
(b) Show that ζ′(s) =− ∑ ∞
n=1 (log n)n−s for σ> 1.
(c) Show that lim σ→∞ ζ′(σ) = 0.
(d) Show that there is no half-plane in which 1 /ζ′(s) can be written as a
convergent Dirichlet series.
6. Let α(s) = ∑ an n−s be a Dirichlet series with an ≥ 0 for all n. Show that
σc = σa , and that
sup
t
|α(s)|= α(σ)
for any given σ>σ c .
7. (V ivanti 1893; Pringsheim 1894) Suppose that f (z) = ∑ ∞
n=0 an zn has radius
of convergence 1 and that an ≥ 0 for all n. Show that z = 1 is a singular point
of f .
8. (Bohr 1910, p. 32) Let t1 = 4, tr +1 = 2tr for r ≥ 1. Put α(s) = ∑ an n−s
where an = 0 unless n ∈ [tr ,2tr ] for some r , in which case put
an =
⎧
⎪
⎪
⎨
⎪
⎪
⎩
t it r
r (n = tr ),
nit r − (n − 1)it r (tr < n < 2tr ),
−(2tr − 1)it r (n = 2tr ).
(a) Show that ∑ 2tr
tr an = 0. | {
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1.3 Euler products and the zeta function 19
(b) Show that if tr ≤ x < 2tr for some r , then A(x ) = [x ]it r where A(x ) =∑
n≤x an .
(c) Show that A(x ) ≪ 1 uniformly for x ≥ 1.
(d) Deduce that α(s) converges for σ> 0.
(e) Show that α(it ) does not converge; conclude that σc = 0.
(f) Show that if σ> 0, then
α(s) =
R∑
r =1
2tr∑
n=tr
an n−s + s
∫ ∞
tR+1
A(x )x −s−1 dx .
(g) Suppose that σ> 0. Show that the above is
2tR∑
n=tR
an n−s + O
(
tR−1
)
+ O
(|s|
σt σ
R+1
)
.
(h) Show that if σ> 0, then
2tR∑
n=tR
an n−s = s
∫ 2tR
tR
[x ]it R x −s−1 dx .
(i) Show that if n ≤ x < n + 1, then ℜ(nit R x −it R ) ≥ 1/2. Deduce that
⏐
⏐
⏐
⏐
∫ 2tR
tR
[x ]it R x −σ−it R −1 dx
⏐
⏐
⏐
⏐≫ t −σ
R .
(j) Suppose that δ> 0 is fixed. Conclude that if R ≥ R0 (δ), then |α(σ +
it R )|≫ t 1−σ
R uniformly for δ ≤ σ ≤ 1 − δ.
(k) Show that ∑ |an |n−σ < ∞ when σ> 1. Deduce that σa = 1.
1.3 Euler products and the zeta function
The situation regarding products of Dirichlet series is somewhat complicated,
but it is useful to note that the formal calculation in (2) is justified if the series
are absolutely convergent.
Theorem 1.8Let α(s) = ∑ an n−s and β(s) = ∑ bn n−s be two Dirichlet se-
ries, and put γ(s) = ∑ cn n−s where the c n are given by (1.3). If s is a point at
which the two series α(s) and β(s) are both absolutely convergent, then γ(s) is
absolutely convergent and γ(s) = α(s)β(s).
The mere convergence of α(s) and β(s) is not sufficient to justify (1.2).
Indeed, the square of the series (1.12) can be shown to have abscissa of conver-
gence≥ 1/4. | {
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20 Dirichlet series: I
A function is called an arithmetic function if its domain is the set Z of inte-
gers, or some subset of the integers such as the natural numbers. An arithmetic
functionf (n) is said to be multiplicative if f (1) = 1 and if f (mn ) = f (m) f (n)
whenever ( m,n) = 1. Also, an arithmetic function f (n) is called totally multi-
plicative if f (1) = 1 and if f (mn ) = f (m) f (n) for all m and n.I f f is multi-
plicative then the Dirichlet series ∑ f (n)n−s factors into a product over primes.
T o see why this is so, we first argue formally (i.e., we ignore questions of con-
vergence). When the product
∏
p
(1 + f ( p) p−s + f ( p2 ) p−2s + f ( p3 ) p−3s +··· )
is expanded, the generic term is
f
(
pk1
1
)
f
(
pk2
2
)
··· f
(
pkr
r
)
(
pk1
1 pk2
2 ··· pkr
r
)s .
Set n = pk1
1 pk2
2 ··· pkr
r . Since f is multiplicative, the above is f (n)n−s . More-
over, this correspondence between products of prime powers and positive inte-
gersn is one-to-one, in view of the fundamental theorem of arithmetic. Hence
after rearranging the terms, we obtain the sum ∑ f (n)n−s . That is, we expect
that
∞∑
n=1
f (n)n−s =
∏
p
(1 + f ( p) p−s + f ( p2 ) p−2s +··· ). (1.14)
The product on the right-hand side is called the Euler product of the Dirichlet
series. The mere convergence of the series on the left does not imply that the
product converges; as in the case of the identity (1.2), we justify (1.14) only
under the stronger assumption of absolute convergence.
Theorem 1.9If f is multiplicative and ∑ | f (n)|n−σ < ∞, then (1.14) holds.
If f is totally multiplicative, then the terms on the right-hand side in (1.14)
form a geometric progression, in which case the identity may be written more
concisely ,
∞∑
n=1
f (n)n−s =
∏
p
(1 − f ( p) p−s )−1 . (1.15)
Proof For any prime p,
∞∑
k=0
| f ( pk )| p−kσ ≤
∞∑
n=1
| f (n)|n−σ < ∞, | {
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1.3 Euler products and the zeta function 21
so each sum on the right-hand side of (1.14) is absolutely convergent. Let
y be a positive real number, and let N be the set of those positive integers
composed entirely of primes not exceeding y, N ={ n : p|n ⇒ p ≤ y}. (Note
that 1 ∈ N.) Since a product of finitely many absolutely convergent series may
be arbitrarily rearranged, we see that
/Pi1 y =
∏
p≤y
(
1 + f ( p) p−s + f ( p2 ) p−2s +···
)
=
∑
n∈N
f (n)n−s .
Hence
⏐
⏐
⏐
⏐/Pi1 y −
∞∑
n=1
f (n)n−s
⏐
⏐
⏐
⏐≤
∑
n /∈N
| f (n)|n−σ.
If n ≤ y then all prime factors of n are ≤ y, and hence n ∈ N. Consequently
the sum on the right above is
≤
∑
n>y
| f (n)|n−σ,
which is small if y is large. Thus the partial products /Pi1 y tend to ∑ f (n)n−s as
y →∞ . □
Let ω(n) denote the number of distinct primes dividing n, and let /Omega1 (n)b e
the number of distinct prime powers dividing n. That is,
ω(n) =
∑
p|n
1,/Omega1 (n) =
∑
pk |n
1 =
∑
pk ∥ n
k. (1.16)
It is easy to distinguish these functions, since ω(n) ≤ /Omega1 (n) for all n, with equal-
ity if and only if n is square-free. These functions are examples of additive
functions because they satisfy the functional relation f (mn ) = f (m) + f (n)
whenever ( m,n) = 1. Moreover, /Omega1 (n)i s totally additive because this func-
tional relation holds for all pairs m,n. An exponential of an additive function is
a multiplicative function. In particular, the Liouville lambda function is the to-
tally multiplicative function λ(n) = (−1)/Omega1 (n) . Closely related is the M ¨obius mu
function, which is defined to be µ(n) = (−1)ω(n) if n is square-free, µ(n) = 0
otherwise. By the fundamental theorem of arithmetic we know that a multi-
plicative (or additive) function is uniquely determined by its values at prime
powers, and similarly that a totally multiplicative (or totally additive) function
is uniquely determined by its values at the primes. Thusµ(n) is the unique
multiplicative function that takes the value −1 at every prime, and the value 0
at every higher power of a prime, while λ(n) is the unique totally multiplicative
function that takes the value −1 at every prime. By using Theorem 1.9 we can | {
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22 Dirichlet series: I
determine the Dirichlet series generating functions of λ(n) and of µ(n) in terms
of the Riemann zeta function.
Corollary 1.10Fo r σ> 1,
∞∑
n=1
n−s = ζ(s) =
∏
p
(1 − p−s )−1 , (1.17)
∞∑
n=1
µ(n)n−s = 1
ζ(s) =
∏
p
(1 − p−s ), (1.18)
and
∞∑
n=1
λ(n)n−s = ζ(2s)
ζ(s) =
∏
p
(1 + p−s )−1 . (1.19)
Proof All three series are absolutely convergent, since ∑ n−σ < ∞ for σ>
1, by the integral test. Since the coefficients are multiplicative, the Euler product
formulae follow by Theorem 1.9. In the first and third cases use the variant
(1.15). On comparing the Euler products in (1.17) and (1.18), it is immediate
that the second of these Dirichlet series is 1/ζ(s). As for (1.19), from the identity
1 + z = (1 − z2 )/(1 − z) we deduce that
∏
p
(1 + p−s ) =
∏
p (1 − p−2s )
∏
p (1 − p−s ) = ζ(s)
ζ(2s) .
□
The manipulation of Euler products, as exemplified above, provides a pow-
erful tool for relating one Dirichlet series to another.
In (1.17) we have expressed ζ(s) as an absolutely convergent product; hence
in particular ζ(s) ̸=0 for σ> 1. W e have not yet defined the zeta function
outside this half-plane, but we shall do so shortly , and later we shall find that
the zeta function does have zeros in the half-planeσ ≤ 1. These zeros play an
important role in determining the distribution of prime numbers.
Many important relations involving arithmetic functions can be expressed
succinctly in terms of Dirichlet series. For example, the fundamental elementary
identity
∑
d |n
µ(d ) =
{
1i f n = 1,
0i f n > 1.
(1.20)
is equivalent to the identity
ζ(s) · 1
ζ(s) = 1, | {
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1.3 Euler products and the zeta function 23
in view of (1.3), (1.17), (1.18), and Theorem 1.6. More generally , if
F (n) =
∑
d |n
f (d ) (1.21)
for all n, then, apart from questions of convergence,
∑
F (n)n−s = ζ(s)
∑
f (n)n−s .
By M ¨ obius inversion, the identity (1.21) is equivalent to the relation
f (n) =
∑
d |n
µ(d ) F (n/d ),
which is to say that
∑
f (n)n−s = 1
ζ(s)
∑
F (n)n−s .
Such formal manipulations can be used to suggest (or establish) many useful
elementary identities.
For σ> 1 the product (1.17) is absolutely convergent. Since log(1 − z)−1 =∑ ∞
k=1 zk /k for |z| < 1, it follows that
log ζ(s) =
∑
p
log(1 − p−s )−1 =
∑
p
∞∑
k=1
k−1 p−ks .
On differentiating, we find also that
ζ′(s)
ζ(s) =−
∑
p
∞∑
k=1
(log p) p−ks
for σ> 1. This is a Dirichlet series, whose nth coefficient is the von Mangoldt
lambda function: /Lambda1 (n) = log p if n is a power of p, /Lambda1 (n) = 0 otherwise.
Corollary 1.11 Fo r σ> 1,
log ζ(s) =
∞∑
n=1
/Lambda1 (n)
log n n−s
and
− ζ′(s)
ζ(s) =
∞∑
n=1
/Lambda1 (n)n−s .
The quotient f ′(s)/f (s), obtained by differentiating the logarithm of f (s),
is known as the logarithmic derivative of f . Subsequently we shall often write
it more concisely as f ′
f (s). | {
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24 Dirichlet series: I
The important elementary identity
∑
d |n
/Lambda1 (d ) = log n (1.22)
is reflected in the relation
ζ(s)
(
− ζ′
ζ (s)
)
=− ζ′(s),
since
−ζ′(s) =
∞∑
n=1
(log n)n−s
for σ> 1.
W e now continue the zeta function beyond the half-plane in which it was
initially defined.
Theorem 1.12Suppose that σ> 0,x > 0, and that s ̸=1. Then
ζ(s) =
∑
n≤x
n−s + x 1−s
s − 1 + {x }
x s − s
∫ ∞
x
{u}u−s−1 du . (1.23)
Here {u} denotes the fractional part of u, so that {u}= u − [u] where [ u]
denotes the integral part of u.
Proof of Theorem 1.12 For σ> 1w eh a v e
ζ(s) =
∞∑
n=1
n−s =
∑
n≤x
n−s +
∑
n>x
n−s .
This second sum we write as
∫ ∞
x
u−s d [u] =
∫ ∞
x
u−s du −
∫ ∞
x
u−s d {u}.
W e evaluate the first integral on the right-hand side, and integrate the second
one by parts. Thus the above is
= x 1−s
s − 1 +{ x }x −s +
∫ ∞
x
{u} du −s .
Since ( u−s )′ =− su −s−1 , the desired formula now follows by Theorem A.3.
The integral in (1.23) is convergent in the half-plane σ> 0, and uniformly so
for σ ≥ δ> 0. Since the integrand is an analytic function of s, it follows that the
integral is itself an analytic function for σ> 0. By the uniqueness of analytic
continuation the formula (1.23) holds in this larger half-plane. □ | {
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1.3 Euler products and the zeta function 25
–10
–
–
–
–
0
10
1 5
Figure 1.2 The Riemann zeta function ζ(s) for 0 < s ≤ 5.
By taking x = 1 in (1.23) we obtain in particular the identity
ζ(s) = s
s − 1 − s
∫ ∞
1
{u}u−s−1 du (1.24)
for σ> 0. Hence we have
Corollary 1.13 The Riemann zeta function has a simple pole at s = 1 with
residue 1, but is otherwise analytic in the half-plane σ> 0.
A graph of ζ(s) that exhibits the pole at s = 1 is provided in Figure 1.2. By
repeatedly integrating by parts we can continue ζ(s) into successively larger
half-planes; this is systematized by using the Euler–Maclaurin summation for-
mula (see Theorem B.5). In Chapter 10 we shall continue the zeta function by a
different method. For the present we note that (1.24) yields useful inequalities
for the zeta function on the real line.
Corollary 1.14The inequalities
1
σ − 1 <ζ (σ) < σ
σ − 1
hold for all σ> 0. In particular , ζ(σ) < 0 for 0 <σ< 1.
Proof From the inequalities 0 ≤{ u} < 1 it follows that
0 ≤
∫ ∞
1
{u}u−σ−1 du <
∫ ∞
1
u−σ−1 du = 1
σ .
This suffices. □ | {
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26 Dirichlet series: I
W e now put the parameter x in (1.23) to good use.
Corollary 1.15 Let δ be fixed, δ> 0. Then for σ ≥ δ,s ̸=1,
∑
n≤x
n−s = x 1−s
1 − s + ζ(s) + O (τx −σ). (1.25)
In addition,
∑
n≤x
1
n = log x + C0 + O (1/x ) (1.26)
where C 0 is Euler’s constant,
C0 = 1 −
∫ ∞
1
{u}u−2 du = 0.5772156649 .... (1.27)
Proof The first estimate follows by crudely estimating the integral in (1.23):
∫ ∞
x
{u}u−s−1 du ≪
∫ ∞
x
u−σ−1 du = x −σ
σ .
As for the second estimate, we note that the sum is
∫ x
1−
u−1 d [u] =
∫ x
1−
u−1 du −
∫ x
1−
u−1 d {u}
= log x + 1 −{ x }/x −
∫ x
1
{u}u−2 du .
The result now follows by writing
∫x
1 =
∫∞
1 −
∫∞
x , and noting that
∫ ∞
x
{u}u−2 du ≪
∫ ∞
x
u−2 du = 1/x .
□
By letting s → 1 in (1.25) and comparing the result with (1.26), or by letting
s → 1 in (1.24) and comparing the result with (1.27), we obtain
Corollary 1.16 Let
ζ(s) = 1
s − 1 +
∞∑
k=0
ak (s − 1)k (1.28)
be the Laurent expansion of ζ(s) at s = 1. Then a 0 is Euler’s constant, a 0 = C0 .
Euler’s constant also arises in the theory of the gamma function. (See
Appendix C and Chapter 10.)
Corollary 1.17Let δ> 0 be fixed. Then
ζ(s) = 1
s − 1 + O (1) | {
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1.3 Euler products and the zeta function 27
uniformly for s in the rectangle δ ≤ σ ≤ 2, |t |≤ 1, and
ζ(s) ≪ (1 + τ1−σ) min
( 1
|σ − 1| , log τ
)
uniformly for δ ≤ σ ≤ 2, |t |≥ 1.
Proof The first assertion is clear from (1.24). When |t | is larger, we obtain
a bound for |ζ(s)| by estimating the sum in (1.25). Assume that x ≥ 2. W e
observe that
∑
n≤x
n−s ≪
∑
n≤x
n−σ ≪ 1 +
∫ x
1
u−σ du
uniformly for σ ≥ 0. If 0 ≤ σ ≤ 1 − 1/log x , then this integral is
(x 1−σ − 1)/(1 − σ) < x 1−σ/(1 − σ). If |σ − 1|≤ 1/log x , then u−σ ≍ u−1
uniformly for 1 ≤ u ≤ x , and hence the integral is ≍
∫x
1 u−1 du = log x .I f
σ ≥ 1 + 1/log x , then the integral is <
∫∞
1 u−σ du = 1/(σ − 1). Thus
∑
n≤x
n−s ≪ (1 + x 1−σ) min
( 1
|σ − 1| , log x
)
(1.29)
uniformly for 0 ≤ σ ≤ 2. The second assertion now follows by taking x = τ
in (1.25). □
1.3.1 Exercises
1. Suppose that f (mn ) = f (m) f (n) whenever ( m,n) = 1, and that f is not
identically 0. Deduce that f (1) = 1, and hence that f is multiplicative.
2. (Stieltjes 1887) Suppose that ∑ an converges, that ∑ |bn | < ∞, and that
cn is given by (1.3). Show that ∑ cn converges to ( ∑ an )(∑ bn ). (Hint:
Write ∑
n≤x cn = ∑
n≤x bn A(x /n) where A( y) = ∑
n≤y an .)
3. Determine ∑ ϕ(n)n−s , ∑ σ(n)n−s , and ∑ |µ(n)|n−s in terms of the zeta
function. Here ϕ(n) is Euler’s ‘totient function’, which is the number of a,
1 ≤ a ≤ n, such that ( a,n) = 1.
4. Let q be a positive integer. Show that if σ> 1, then
∞∑
n=1
(n,q )=1
n−s = ζ(s)
∏
p|q
(1 − p−s ).
5. Show that if σ> 1, then
∞∑
n=1
d (n)2 n−s = ζ(s)4/ζ(2s). | {
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28 Dirichlet series: I
6. Let σa (n) = ∑
d |n d a . Show that
∞∑
n=1
σa (n)σb (n)n−s = ζ(s)ζ(s − a)ζ(s − b)ζ(s − a − b)/ζ(2s − a − b)
when σ> max (1,1 +ℜ a,1 +ℜ b,1 +ℜ (a + b)).
7. Let F (s) = ∑
p (log p) p−s , G (s) = ∑
p p−s for σ> 1. Show that in this
half-plane,
− ζ′
ζ (s) =
∞∑
k=1
F (ks ),
F (s) =−
∞∑
d =1
µ(d ) ζ′
ζ (ds ),
log ζ(s) =
∞∑
k=1
G (ks )/k,
G (s) =
∞∑
d =1
µ(d )
d log ζ(ds ).
8. Let F (s) and G (s) be defined as in the preceding problem. Show that if
σ> 1, then
∞∑
n=1
ω(n)n−s = ζ(s)G (s) = ζ(s)
∞∑
d =1
µ(d )
d log ζ(ds ),
∞∑
n=1
/Omega1 (n)n−s = ζ(s)
∞∑
k=1
G (ks ) = ζ(s)
∞∑
k=1
ϕ(k)
k log ζ(ks ).
9. Let t be a fixed real number, t ̸=0. Describe the limit points of the sequence
of partial sums ∑
n≤x n−1−it .
10. Show that ∑ N
n=1 n−1 > log N + C0 for all positive integers N , and that∑
n≤x n−1 > log x for all positive real numbers x .
11. (a) Show that if an is totally multiplicative, and if α(s) = ∑ an n−s has
abscissa of convergence σc , then
∞∑
n=1
(−1)n−1 an n−s = (1 − 2a2 2−s )α(s)
for σ>σ c .
(b) Show that
∞∑
n=1
(−1)n−1 n−s = (1 − 21−s )ζ(s)
for σ> 0. | {
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1.3 Euler products and the zeta function 29
(c) (Shafer 1984) Show that
∞∑
n=1
(−1)n (log n)n−1 = C0 log 2 − 1
2 (log 2) 2 .
12. (Stieltjes 1885) Show that if k is a positive integer, then
∑
n≤x
(log n)k
n = (log x )k+1
k + 1 + Ck + Ok
((log x )k
x
)
for x ≥ 1 where
Ck =
∫ ∞
1
{u}(log u)k−1 (k − log u)u−2 du .
Show that the numbers ak in (1.28) are given by ak = (−1)k Ck /k!.
13. Let D be the disc of radius 1 and centre 2. Suppose that the numbers εk tend
monotonically to 0, that the numbers tk tend monotonically to 0, and that
the numbers Nk tend monotonically to infinity . W e consider the Dirichlet
series α(s) = ∑
n an n−s with coefficients an = εk nit k for Nk−1 < n ≤ Nk .
For suitable choices of the εk , tk , and Nk we show that the series converges
at s = 1 but that it is not uniformly convergent in D.
(a) Suppose that σk = 2 −
√
1 − t 2
k , so that sk = σk + it k ∈ D. Show that if
N
t 2
k
k ≪ 1, (1.30)
then
⏐
⏐
⏐
∑
Nk−1 <n≤Nk
an n−sk
⏐
⏐
⏐≫ εk log Nk
Nk−1
.
Thus if
εk log Nk
Nk−1
≫ 1 (1.31)
then the series is not uniformly convergent in D.
(b) By using Corollary 1.15, or otherwise, show that if ( a,b] ⊆ ( Nk−1 ,Nk ],
then
∑
a<n≤b
an n−1 ≪ εk
tk
.
Hence if
∞∑
k=1
εk
tk
< ∞, (1.32)
then the series α(1) converges. | {
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30 Dirichlet series: I
(c) Show that the parameters can be chosen so that (1.30)–(1.32) hold, say
by taking Nk = exp(1/εk ) and tk = ε1/2
k with εk tending rapidly to 0.
14. Let t (n) = (−1)/Omega1 (n)−ω(n) ∏
p|n ( p − 1)−1 , and put T (s) = ∑
n t (n)n−s .
(a) Show that for σ> 0, T (s) has the absolutely convergent Euler product
T (s) =
∏
p
(
1 + 1
( p − 1)( ps + 1)
)
.
(b) Determine all zeros of the function 1 + 1/(( p − 1)( ps + 1)).
(c) Show that the line σ = 0 is a natural boundary of the function T (s).
15. Suppose throughout that 0 <α ≤ 1. For σ> 1 we define the Hurwitz zeta
function by the formula
ζ(s,α) =
∞∑
n=0
(n + α)−s .
Thus ζ(s,1) = ζ(s).
(a) Show that ζ(s,1/2) = (2s − 1)ζ(s).
(b) Show that if x ≥ 0 then
ζ(s,α) =
∑
0≤n≤x
(n + α)−s + (x + α)1−s
s − 1 + {x }
(x + α)s
− s
∫ ∞
x
{u}(u + α)−s−1 du .
(c) Deduce that ζ(s,α) is an analytic function of s for σ> 0 apart from a
simple pole at s = 1 with residue 1.
(d) Show that
lim
s→1
(
ζ(s,α) − 1
s − 1
)
= 1/α − log α −
∫ ∞
0
{u}
(u + α)2 du .
(e) Show that
lim
s→1
(
ζ(s,α) − 1
s − 1
)
=
∑
0≤n≤x
1
n + α − log(x + α) + {x }
x + α
−
∫ ∞
x
{u}
(u + α)2 du .
(f) Let x →∞ in the above, and use (C.2), (C.10) to show that
lim
s→1
(
ζ(s,α) − 1
s − 1
)
=− Ŵ′
Ŵ (α).
(This is consistent with Corollary 1.16, in view of (C.11).) | {
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1.4 Notes 31
1.4 Notes
Section 1.1. For a brief introduction to the Hardy–Littlewood circle method,
including its application to W aring’s problem, see Davenport (2005). For a
comprehensive account of the method, see V aughan (1997). Other examples
of the fruitful use of generating functions are found in many sources, such as
Andrews (1976) and Wilf (1994).
Algorithms for the efficient computation of π(x ) have been developed
by Meissel (Lehmer, 1959), Mapes (1963), Lagarias, Miller & Odlyzko
(1985), Del´ eglise & Rivat (1996), and by X. Gourdon. For discussion
of these methods, see Chapter 1 of Riesel (1994) and the web page of
Gourdon & Sebah at http://numbers.computation.free.fr/Constants/Primes/
countingPrimes.html.
The ‘big oh’ notation was introduced by Paul Bachmann (1894, p. 401). The
‘little oh’ was introduced by Edmund Landau (1909a, p. 61). The ≍ notation
was introduced by Hardy (1910, p. 2). Our notation f ∼ g also follows Hardy
(1910). The Omega notation was introduced by G. H. Hardy and J. E. Littlewood
(1914, p. 225). Ingham (1932) replaced the/Omega1 R and /Omega1 L of Hardy and Littlewood
by /Omega1 + and /Omega1 −. The ≪ notation is due to I. M. V inogradov .
Section 1.2. The series ∑ an n−s is called an ordinary Dirichlet series,
to distinguish it from a generalized Dirichlet series, which is a sum of the
form ∑ an e−λn s where 0 <λ 1 <λ 2 < ··· , λn →∞ . W e see that generalized
Dirichlet series include both ordinary Dirichlet series ( λn = log n) and power
series ( λn = n). Theorems 1.1, 1.3, 1.6, and 1.7 extend naturally to generalized
Dirichlet series, and even to the more general class of functions
∫∞
0 e−us dA (u)
where A(u) is assumed to have finite variation on each finite interval [0 ,U ].
The proof of the general form of Theorem 1.6 must be modified to depend on
uniform, rather than absolute, convergence, since a generalized Dirichlet series
may be never more than conditionally convergent (e.g.,∑ (−1)n (log n)−s ).
If we put a = lim sup(log n)/λn , then the general form of Theorem 1.4
reads σc ≤ σa ≤ σc + a. Hardy & Riesz (1915) have given a detailed ac-
count of this subject, with historical attributions. See also Bohr & Cram´ er
(1923).
Jensen (1884) showed that the domain of convergence of a generalized
Dirichlet series is always a half-plane. The more precise information provided
by Theorem 1.1 is due to Cahen (1894) who proved it not only for ordinary
Dirichlet series but also for generalized Dirichlet series.
The construction in Exercise 1.2.8 would succeed with the simpler choice
an = nit r for tr ≤ n ≤ 2tr , an = 0 otherwise, but then to complete the argu-
ment one would need a further tool, such as the Kusmin–Landau inequality | {
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32 Dirichlet series: I
(cf. Mordell 1958). The square of the Dirichlet series in Exercise 1.2.8 has ab-
scissa of convergence 1/2; this bears on the result of Exercise 2.1.9. Information
concerning the convergence of the product of two Dirichlet series is found in
Exercises 1.3.2, 2.1.9, 5.2.16, and in Hardy & Riesz (1915).
Theorem 1.7 originates in Landau (1905). The analogue for power series had
been proved earlier by V ivanti (1893) and Pringsheim (1894). Landau’s proof
extends to generalized Dirichlet series (including power series).
Section 1.3. The hypothesis ∑ | f (n)|n−σ < ∞ of Theorem 1.9 is equivalent
to the assertion that
∏
p
(1 +| f ( p)| p−σ +| f ( p2 )| p−2σ +··· ) < ∞,
which is slightly stronger than merely asserting that the Euler product converges
absolutely . W e recall that a product∏
n (1 + an ) is said to be absolutely con-
vergent if ∏
n (1 +| an |) < ∞. T o see that the hypothesis ∏
p (1 +| f ( p) p−s +
···| ) < ∞ is not sufficient, consider the following example due to Ingham:
For every prime p we take f ( p) = 1, f ( p2 ) =− 1, and f ( pk ) = 0 for k > 2.
Then the product is absolutely convergent at s = 0, but the terms f (n) do not
tend to 0, and hence the series ∑ f (n) diverges. Indeed, it can be shown that∑
n≤x f (n) ∼ cx as x →∞ where c = ∏
p
(
1 − 2 p−2 + p−3 )
> 0.
Euler (1735) defined the constant C0 , which he denoted C .
Mascheroni (1790) called the constant γ, which is in common use, but
we wish to reserve this symbol for the imaginary part of a zero of the
zeta function or anL -function. It is conjectured that Euler’s constant C0
is irrational. The early history of the determination of the initial digits of
C0 has been recounted by Nielsen (1906, pp. 8–9). More recently , Wrench
(1952) computed 328 digits, Knuth (1963) computed 1,271 digits, Sweeney
(1963) computed 3,566 digits, Beyer & W aterman (1974) computed 4,879
digits, Brent (1977) computed 20,700 digits, Brent & McMillan (1980)
computed 30,100 digits. At this time, it seems that more than 108 digits
have been computed – see the web page of X. Gourdon & P . Sebah at
http://numbers.computation.free.fr/Constants/Gamma/gamma.html. T o 50
places, Euler’s constant is
C0 = 0.57721 56649 01532 86060 65120 90082 40243 10421 59335 93992 .
Statistical analysis of the continued fraction coefficients of C0 suggest that it
satisfies the Gauss–Kusmin law , which is to say that C0 seems to be a typical
irrational number.
Landau & W alfisz (1920) showed that the functions F (s) and G (s) of Ex-
ercise 1.3.7 have the imaginary axis σ = 0 as a natural boundary . For further | {
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1.5 References 33
work on Dirichlet series with natural boundaries see Estermann (1928a,b) and
Kurokawa (1987).
1.5 References
Andrews, G. E. (1976). The Theory of P artitions , Reprint. Cambridge: Cambridge Uni-
versity Press (1998).
Bachmann, P . (1894). Zahlentheorie, II, Die analytische Zahlentheorie , Leipzig:
T eubner.
Beyer, W . A. & W aterman, M. S. (1974). Error analysis of a computation of Euler’s
constant and ln 2, Math. Comp. 28, 599–604.
Bohr, H. (1910). Bidrag til de Dirichlet’ske Rækkers theori , København: G. E. C. Gad;
Collected Mathematical W orks , V ol. I, København: Danske Mat. Forening, 1952.
A3.
Bohr, H. & Cram´ er, H. (1923). Die neuere Entwicklung der analytischen Zahlentheo-
rie, Enzyklop¨ adie der Mathematischen Wissenschaften, 2, C8, 722–849; H. Bohr,
Collected Mathematical W orks , V ol. III, København: Dansk Mat. Forening, 1952,
H; H. Cram´ er, Collected W orks , V ol. 1, Berlin: Springer-V erlag, 1952, pp. 289–
416.
Brent, R. P . (1977). Computation of the regular continued fraction of Euler’s constant,
Math. Comp. 31, 771–777.
Brent, R. P . & McMillan, E. M. (1980). Some new algorithms for high-speed computation
of Euler’s constant, Math. Comp. 34, 305–312.
Cahen, E. (1894). Sur la fonction ζ(s) de Riemann et sur des fonctions analogues, Ann.
de l’ ´Ecole Normale (3) 11, 75–164.
Davenport, H. (2005). Analytic Methods for Diophantine Equations and Diophantine
Inequalities. Second edition, Cambridge: Cambridge University Press.
Del´ eglise, M. & Rivat, J. (1996). Computing π(x ): the Meissel, Lehmer, Lagarias, Miller,
Odlyzko method, Math. Comp. 65, 235–245.
Estermann, T . (1928a). On certain functions represented by Dirichlet series, Proc. Lon-
don Math. Soc. (2) 27, 435–448.
(1928b). On a problem of analytic continuation, Proc. London Math. Soc. (2) 27,
471–482.
Euler, L. (1735). De Progressionibus harmonicus observationes, Comm. Acad. Sci. Imper .
P etropol.7, 157; Opera Omnia, ser. 1, vol. 14, T eubner, 1914, pp. 93–95.
Hardy , G. H. (1910). Orders of Infinity . Cambridge Tract 12, Cambridge: Cambridge
University Press.
Hardy , G. H. & Littlewood, J. E. (1914). Some problems of Diophantine approximation
(II), Acta Math. 37, 193–238; Collected P apers, V ol I. Oxford: Oxford University
Press. 1966, pp. 67–112.
Hardy , G. H. & Riesz, M. (1915). The General Theory of Dirichlet’s Series , Cambridge
Tract No. 18. Cambridge: Cambridge University Press. Reprint: Stechert–Hafner
(1964).
Ingham, A. E. (1932). The Distribution of Prime Numbers , Cambridge Tract 30. Cam-
bridge: Cambridge University Press. | {
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34 Dirichlet series: I
Jensen, J. L. W . V . (1884). Om Rækkers Konvergens, Tidsskrift for Math. (5) 2, 63–72.
(1887). Sur la fonction ζ(s) de Riemann, Comptes Rendus Acad. Sci. Paris 104,
1156–1159.
Knuth, D. E. (1962). Euler’s constant to 1271 places, Math. Comp. 16, 275–281.
Kurokawa, N. (1987). On certain Euler products, Acta Arith. 48, 49–52.
Lagarias, J. C., Miller, V . S., & Odlyzko, A. M. (1985). Computing π(x ): The Meissel–
Lehmer method, Math. Comp. 44, 537–560.
Lagarias, J. C. & Odlyzko, A. M. (1987). Computing π(x ): An analytic method, J.
Algorithms 8, 173–191.
Landau, E. (1905). ¨Uber einen Satz von Tschebyschef, Math. Ann. 61, 527–550;
Collected W orks, V ol. 2, Essen: Thales, 1986, pp. 206–229.
(1909a). Handbuch der Lehre von der V erteilung der Primzahlen , Leipzig: T eubner.
Reprint: Chelsea (1953).
(1909b). ¨Uber das Konvergenzproblem der Dirichlet’schen Reihen, Rend. Circ. Mat.
P alermo 28, 113–151; Collected W orks, V ol. 4, Essen: Thales, 1986, pp. 181–220.
Landau, E. & W alfisz, A. (1920). ¨Uber die Nichtfortsetzbarkeit einiger durch Dirich-
letsche Reihen definierte Funktionen, Rend. Circ. Mat. P alermo 44, 82–86;
Collected W orks, V ol. 7, Essen: Thales, 1986, pp. 252–256.
Lehmer, D. H. (1959). On the exact number of primes less than a given limit, Illinois J.
Math. 3, 381–388.
Mapes, D. C. (1963). Fast method for computing the number of primes less than a given
limit, Math. Comp. 17, 179–185.
Mascheroni, L. (1790). Abnotationes ad calculum integrale Euleri , V ol. 1. Ticino:
Galeatii. Reprinted in the Opera Omnia of L. Euler, Ser. 1, V ol 12, T eubner, 1914,
pp. 415–542.
Mordell, L. J. (1958). On the Kusmin–Landau inequality for exponential sums, Acta
Arith. 4, 3–9.
Nielsen, N. (1906). Handbuch der Theorie der Gammafunktion . Leipzig: T eubner.
Pringsheim, A. (1894). ¨Uber Functionen, welche in gewissen Punkten endliche Differen-
tialquotienten jeder endlichen Ordnung, aber kein T aylorsche Reihenentwickelung
besitzen,Math. Ann. 44, 41–56.
Riesel, H. (1994). Prime Numbers and Computer Methods for F actorization , Second
ed., Progress in Math. 126. Boston: Birkh¨ auser.
Shafer, R. E. (1984). Advanced problem 6456, Amer . Math. Monthly 91, 205.
Stieltjes, T . J. (1885). Letter 75 in Correspondance d’Hermite et de Stieltjes , B. Baillaud
& H. Bourget, eds., Paris: Gauthier-V illars, 1905.
(1887). Note sur la multiplication de deux s´ eries, Nouvelles Annales (3) 6, 210–215.
Sweeney , D. W . (1963). On the computation of Euler’s constant, Math. Comp. 17, 170–
178.
V aughan, R. C. (1997). The Hardy–Littlewood Method , Second edition, Cambridge Tract
125. Cambridge: Cambridge University Press.
V ivanti, G. (1893). Sulle serie di potenze, Rivista di Mat. 3, 111–114.
W agon, S. (1987). Fourteen proofs of a result about tiling a rectangle, Amer . Math.
Monthly 94, 601–617.
Widder, D. V . (1971). An Introduction to Transform Theory . New Y ork: Academic Press.
Wilf, H. (1994). Generatingfunctionology, Second edition. Boston: Academic Press.
Wrench, W . R. Jr (1952). A new calculation of Euler’s constant, MTAC 6, 255. | {
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2
The elementary theory of arithmetic functions
2.1 Mean values
W e say that an arithmetic function F (n) has a mean value c if
lim
N →∞
1
N
N∑
n=1
F (n) = c.
In this section we develop a simple method by which mean values can be shown
to exist in many interesting cases.
If two arithmetic functions f and F are related by the identity
F (n) =
∑
d |n
f (d ), (2.1)
then we can write f in terms of F :
f (n) =
∑
d |n
µ(d ) F (n/d ). (2.2)
This is the M ¨obius inversion formula . Conversely , if (2.2) holds for all n then
so also does (2.1). If f is generally small then F has an asymptotic mean value.
T o see this, observe that
∑
n≤x
F (n) =
∑
n≤x
∑
d |n
f (d ).
By iterating the sums in the reverse order, we see that the above is
=
∑
d ≤x
f (d )
∑
n≤x
d |n
1 =
∑
d ≤x
f (d )[x /d ].
35 | {
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36 The elementary theory of arithmetic functions
Since [ y] = y + O (1), this is
= x
∑
d ≤x
f (d )
d + O
(∑
d ≤x
| f (d )|
)
. (2.3)
Thus F has the mean value ∑ ∞
d=1 f (d )/d if this series converges and if∑
d ≤x | f (d )|= o(x ). This approach, though somewhat crude, often yields use-
ful results.
Theorem 2.1Let ϕ(n) be Euler’s totient function. Then for x ≥ 2,
∑
n≤x
ϕ(n)
n = 6
π2 x + O (log x ).
Proof W e recall that ϕ(n) = n ∏
p|n (1 − 1/p). On multiplying out the prod-
uct, we see that
ϕ(n)
n =
∑
d |n
µ(d )
d .
On taking f (d ) = µ(d )/d in (2.3), it follows that
∑
n≤x
ϕ(n)
n = x
∑
d ≤x
µ(d )
d 2 + O (log x ).
Since ∑
d >x d −2 ≪ x −1 , we see that
∑
d ≤x
µ(d )
d 2 =
∞∑
d =1
µ(d )
d 2 + O
(1
x
)
= 1
ζ(2) + O
(1
x
)
by Corollary 1.10. From Corollary B.3 we know that ζ(2) = π2 /6; hence the
proof is complete. □
Let Q(x ) denote the number of square-free integers not exceeding x , Q(x ) =∑
n≤x µ(n)2 . W e now calculate the asymptotic density of these numbers.
Theorem 2.2 F or all x ≥ 1,
Q(x ) = 6
π2 x + O
(
x 1/2 )
.
Proof Every positive integer n is uniquely of the form n = ab 2 where a is
square-free. Thus n is square-free if and only if b = 1, so that by (1.20)
∑
d 2 |n
µ(d ) =
∑
d |b
µ(d ) = µ(n)2 . (2.4) | {
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2.1 Mean values 37
This is a relation of the shape (2.1) where f (d ) = µ(
√
d )i f d is a perfect square,
and f (d ) = 0 otherwise. Hence by (2.3),
Q(x ) = x
∑
d 2 ≤x
µ(d )
d 2 + O
(∑
d 2 ≤x
1
)
.
The error term is ≪ x 1/2 , and the sum in the main term is treated as in the
preceding proof. □
W e note that the argument above is routine once the appropriate identity
(2.4) is established. This relation can be discovered by considering (2.2), or by
using Dirichlet series: LetQ denote the class of square-free numbers. Then for
σ> 1,
∑
n∈Q
n−s =
∏
p
(1 + p−s ) =
∏
p
1 − p−2s
1 − p−s = ζ(s)
ζ(2s) .
Now 1 /ζ(2s) can be written as a Dirichlet series in s, with coefficients f (n) =
µ(d )i f n = d 2 , f (n) = 0 otherwise. Hence the convolution equation (2.4) gives
the coefficients of the product Dirichlet series ζ(s) · 1/ζ(2s).
Suppose that ak , bm , cn are joined by the convolution relation
cn =
∑
km =n
ak bm , (2.5)
and that A(x ), B (x ), C (x ) are their respective summatory functions. Then
C (x ) =
∑
km ≤x
ak bm , (2.6)
and it is useful to note that this double sum can be iterated in various ways. On
one hand we see that
C (x ) =
∑
k≤x
ak B (x /k); (2.7)
this is the line of reasoning that led to (2.3) (take ak = f (k), bm = 1). At the
opposite extreme,
C (x ) =
∑
m≤x
bm A(x /m), (2.8)
and between these we have the more general identity
C (x ) =
∑
k≤y
ak B (x /k) +
∑
m≤x /y
bm A(x /m) − A( y) B (x /y) (2.9)
for 0 < y ≤ x . This is obvious once it is observed that the first term on the right
sums those terms ak bm for which km ≤ x , k ≤ y, the second sum includes the | {
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38 The elementary theory of arithmetic functions
pairs ( k,m) for which km ≤ x , m ≤ x /y, and the third term subtracts those ak bm
for which k ≤ y, m ≤ x /y, since these ( k,m) were included in both the previous
terms. The advantage of (2.9) over (2.7) is that the number of terms is reduced
(≪ y + x /y instead of ≪ x ), and at the same time A and B are evaluated only
at large values of the argument, so that asymptotic formulæ for these quantities
may be expected to be more accurate. For example, if we wish to estimate the
average size ofd (n) we take ak = bm = 1, and then from (2.3) we see that
∑
n≤x
d (n) = x log x + O (x ).
T o obtain a more accurate estimate we observe that the first term on the
right-hand side of (2.9) is
∑
k≤y
[x /k] = x
∑
k≤y
1/k + O ( y).
By Corollary 1.15 this is
x log y + C0 x + O (x /y + y).
Here the error term is minimized by taking y = x 1/2 . The second term
on the right in (2.9) is then identical to the first, and the third term is
[x 1/2 ]2 = x + O (x 1/2 ), and we have
Theorem 2.3 Fo r x ≥ 2.
∑
n≤x
d (n) = x log x + (2C0 − 1)x + O
(
x 1/2 )
.
W e often construct estimates with one or more parameters, and then choose
values of the parameters to optimize the result. The instance above is typical –
we minimizedx /y + y by taking y = x 1/2 . Suppose, more generally , that we
wish to minimize T1 ( y) + T2 ( y) where T1 is a decreasing function, and T2 is
an increasing function. W e could differentiate and solve for a root of T ′
1 ( y) +
T ′
2 ( y) = 0, but there is a quicker method: Find y0 so that T1 ( y0 ) = T2 ( y0 ). This
does not necessarily yield the exact minimum value of T1 ( y) + T2 ( y) ,b u ti ti s
easy to see that
T1 ( y0 ) ≤ min
y
(T1 ( y) + T2 ( y)) ≤ 2T1 ( y0 ),
so the bound obtained in this way is at most twice the optimal bound.
Despite the great power of analytic techniques, the ‘method of the hyperbola’
used above is a valuable tool. The sequence cn given by (2.5) is called the
Dirichlet convolution of ak and bm ; in symbols, c = a ∗ b. Arithmetic functions
form a ring when equipped with pointwise addition, ( a + b)n = an + bn , and | {
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2.1 Mean values 39
Dirichlet convolution for multiplication. This ring is called the ring of formal
Dirichlet series . Manipulations of arithmetic functions in this way correspond
to manipulations of Dirichlet series without regard to convergence. This is
analogous to the ring of formal power series, in which multiplication is provided
by Cauchy convolution,cn = ∑
k+m=n ak bm .
In the ring of formal Dirichlet series we let O denote the arithmetic function
that is identically 0; this is the additive identity . The multiplicative identity is i
where i1 = 1, in = 0 for n > 1. The arithmetic function that is identically 1 we
denote by 1, and we similarly abbreviate µ(n), /Lambda1 (n), and log n by µ, Λ , and
L. In this notation, the characteristic property of µ(n) is that µ ∗ 1 = i , which
is to say that µ and 1 are convolution inverses of each other, and the M ¨ obius
inversion formula takes the compact form
a ∗ 1 = b ⇐⇒ a = b ∗ µ.
In the elementary study of prime numbers the relations Λ ∗ 1 = L, L ∗ µ = Λ
are fundamental.
2.1.1 Exercises
1. (de la V all´ ee Poussin 1898; cf. Landau 1911) Show that
∑
n≤x
{x /n}= (1 − C0 )x + O
(
x 1/2 )
where C0 is Euler’s constant, and {u}= u − [u] is the fractional part of u.
2. (Duncan 1965; cf. Rogers 1964, Orr 1969) Let Q(x ) be defined as in The-
orem 2.2.
(a) Show thatQ( N ) ≥ N − ∑
p [ N /p2 ] for every positive integer N .
(b) Justify the relations
∑
p
1
p2 < 1
4 +
∞∑
k=1
1
(2k + 1)2 < 1
4 + 1
2
∞∑
k=1
(1
2k − 1
2k + 2
)
= 1/2.
(c) Show that Q( N ) > N /2 for all positive integers N .
(d) Show that every positive integer n > 1 can be written as a sum of two
square-free numbers.
3. (Linfoot & Evelyn 1929) Let Qk denote the set of positive kth power free
integers (i.e., q ∈ Qk if and only if mk |q ⇒ m = 1).
(a) Show that
∑
n∈Qk
n−s = ζ(s)
ζ(ks )
for σ> 1. | {
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40 The elementary theory of arithmetic functions
(b) Show that for any fixed integer k > 1
∑
n≤x
n∈Qk
1 = x
ζ(k) + O
(
x 1/k )
for x ≥ 1.
4. (cf. Evelyn & Linfoot 1930) Let N be a positive integer, and suppose that
P is square-free.
(a) Show that the number of residue classes n (mod P 2 ) for which ( n,P 2 )
is square-free and ( N − n,P 2 ) is square-free is
P 2 ∏
p| P
p2 |N
(
1 − 1
p2
)∏
p| P
p2 ∤N
(
1 − 2
p2
)
.
(b) Show that the number of integers n,0 < n < N , for which ( n,P 2 )i s
square-free and ( N − n,P 2 ) is square-free is
N
∏
p| P
p2 |N
(
1 − 1
p2
)∏
p| P
p2 ∤N
(
1 − 2
p2
)
+ O ( P 2 ).
(c) Show that the number of n,0 < n < N , such that n is divisible by the
square of a prime > y is ≪ N /y.
(d) T ake P to be the product of all primes not exceeding y. By letting y
tend to infinity slowly , show that the number of ways of writing N as
a sum of two square-free integers is ∼ c( N ) N where
c( N ) = a
∏
p2 |N
(
1 + 1
p2 − 2
)
, a =
∏
p
(
1 − 2
p2
)
.
5. (cf. Hille 1937) Suppose that f (x ) and F (x ) are complex-valued functions
defined on [1 ,∞). Show that
F (x ) =
∑
n≤x
f (x /n)
for all x if and only if
f (x ) =
∑
n≤x
µ(n) F (x /n)
for all x .
6. (cf. Hartman & Wintner 1947) Suppose that ∑ | f (n)|d (n) < ∞, and that∑ |F (n)|d (n) < ∞. Show that
F (n) =
∑
m
n|m
f (m) | {
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2.1 Mean values 41
for all n if and only if
f (n) =
∑
m
n|m
µ(m/n) F (m).
7. (Jarn´ ık 1926; cf. Bombieri & Pila 1989) Let C be a simple closed curve in
the plane, of arc length L . Show that the number of ‘lattice points’ ( m,n),
m,n ∈ Z, lying on C is at most L + 1. Show that if C is strictly convex
then the number of lattice points on C is ≪ 1 + L 2/3 , and that this estimate
is best possible.
8. Let C be a simple closed curve in the plane, of arc length L that encloses
a region of area A. Let N be the number of lattice points inside C . Show
that |N − A|≤ 3(L + 1).
9. Let r (n) be the number of pairs ( j,k) of integers such that j 2 + k2 = n.
Show that
∑
n≤x
r (n) = πx + O
(
x 1/2 )
.
10. (Stieltjes 1887) Suppose that ∑ an , ∑ bn are convergent series, and that
cn = ∑
km =n ak bm . Show that ∑ cn n−1/2 converges. (Hence if two Dirichlet
series have abscissa of convergence ≤ σ then the product series γ(s) =
α(s)β(s) has abscissa of convergence σc ≤ σ + 1/2.)
11. (a) Show that ∑
n≤x ϕ(n) = (3/π2 )x 2 + O (x log x ) for x ≥ 2.
(b) Show that
∑
m≤x
n≤x
(m,n)=1
1 =− 1 + 2
∑
n≤x
ϕ(n)
for x ≥ 1. Deduce that the expression above is (6 /π2 )x 2 + O (x log x ).
12. Let σ(n) = ∑
d |n d . Show that
∑
n≤x
σ(n) = π2
12 x 2 + O (x log x )
for x ≥ 2.
13. (Landau 1900, 1936; cf. Sitaramachandrarao 1982, 1985, Nowak 1989)
(a) Show that n/ϕ(n) = ∑
d |n µ(d )2 /ϕ(d ).
(b) Show that
∑
n≤x
n
ϕ(n) = ζ(2)ζ(3)
ζ(6) x + O (log x )
for x ≥ 2. | {
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42 The elementary theory of arithmetic functions
(c) Show that
∞∑
d =1
µ(d )2 log d
d ϕ(d ) =
(∑
p
log p
p2 − p + 1
)∏
p
(
1 + 1
p( p − 1)
)
.
(d) Show that for x ≥ 2,
∑
n≤x
1
ϕ(n) = ζ(2)ζ(3)
ζ(6)
(
log x +C0 −
∑
p
log p
p2 − p + 1
)
+ O ((log x )/x ).
14. Let κ be a fixed real number. Show that
∑
n≤x
(ϕ(n)
n
)κ
= c(κ)x + O (x ε)
where
c(κ) =
∏
p
(
1 − 1
p (1 − (1 − 1/p)κ)
)
.
15. (cf. Grosswald 1956, Bateman1957)
(a) By using Euler products, or otherwise, show that
2ω(n) =
∑
d 2 m=n
µ(d )d (m).
(b) Deduce that
∑
n≤x
2ω(n) = 6
π2 x log x + cx + O
(
x 1/2 log x
)
for x ≥ 2 where c = 2C0 − 1 − 2ζ′(2)/ζ(2)2 .
(c) Show also that
∑
n≤x
2/Omega1 (n) = Cx (log x )2 + O (x log x )
where
C = 1
8 log 2
∏
p>2
(
1 + 1
p( p − 2)
)
.
16. (a) Show that for any positive integer q ,
∑
d |q
µ(d ) log d
d =− ϕ(q )
q
∑
p|q
log p
p − 1 .
(b) Show that for any real number x ≥ 1 and any positive integer q ,
∑
m≤x
(m,q )=1
1
m =
(
log x + C0 +
∑
p|q
log p
p − 1
)ϕ(q )
q + O
(
2ω(q ) /x
)
. | {
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2.1 Mean values 43
(c) Show that for any real number x ≥ 2 and any positive integer q ,
∑
n≤x
(n,q )=1
1
ϕ(n) = ζ(2)ζ(3)
ζ(6)
∏
p|q
(
1 − p
p2 − p + 1
)(
log x + C0 +
∑
p|q
log p
p − 1
−
∑
p∤q
log p
p2 − p + 1
)
+ O
(
2ω(q ) log x
x
)
.
17. (cf. W ard 1927) Show that for x ≥ 2,
∑
n≤x
µ(n)2
ϕ(n) = log x + C0 +
∑
p
log p
p( p − 1) + O
(
x −1/2 log x
)
.
18. Let dk (n) be the number of ordered k-tuples ( d1 ,..., dk ) of positive integers
such that d1 d2 ··· dk = n.
(a) Show that dk (n) = ∑
d |n dk−1 (d ).
(b) Show that ∑ ∞
n=1 dk (n)n−s = ζ(s)k for σ> 1.
(c) Show that for every fixed positive integer k,
∑
n≤x
dk (n) = xP k (log x ) + O
(
x 1−1/k (log x )k−2 )
for x ≥ 2, where P ∈ R[z] has degree k − 1 and leading coefficient
1/(k − 1)!.
19. (cf. Erd ˝ os & Szekeres 1934, Schmidt 1967/68) Let An denote the number
of non-isomorphic Abelian groups of order n.
(a) Show that ∑ ∞
n=1 An n−s = ∏ ∞
k=1 ζ(ks ) for σ> 1.
(b) Show that
∑
n≤x
An = cx + O
(
x 1/2 )
where c = ∏ ∞
k=2 ζ(k).
20. (Wintner 1944, p. 46) Suppose that ∑
d |g(d )|/d < ∞. Show that∑
d ≤x |g(d )|= o(x ). Suppose also that ∑
n≤x f (n) = cx + o(x ), and put
h(n) = ∑
d |n f (d )g(n/d ). Show that
∑
n≤x
h(n) = cgx + o(x )
where g = ∑
d g(d )/d .
21. (a) Show that if a2 is the largest perfect square ≤ x then x − a2 ≤ 2√
x .
(b) Let a2 be as above, and let b2 be the least perfect square such that a2 +
b2 > x . Show that a2 + b2 < x + 6x 1/4 . Thus for any x ≥ 1, there is
a sum of two squares in the interval ( x ,x + 6x 1/4 ). (It is somewhat | {
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44 The elementary theory of arithmetic functions
embarrassing that this is the best-known upper bound for gaps between
sums of two squares.)
22. (Feller & T ornier 1932) Let f (n) denote the multiplicative function such
that f ( p) = 1 for all p, and f ( pk ) =− 1 whenever k > 1.
(a) Show that
∞∑
n=1
f (n)
ns = ζ(s)
∏
p
(
1 − 2
p2s
)
for σ> 1.
(b) Deduce that
f (n) =
∑
d 2 |n
µ(d )2ω(d ) .
(c) Explain why 2 ω(n) ≤ d (n) for all n.
(d) Show that
∑
n≤x
f (n) = ax + O
(
x 1/2 log x
)
where a is the constant of Exercise 3.
(e) Let g(n) denote the number of primes p such that p2 |n. Show
that the set of n for which g(n) is even has asymptotic density
(1 + a)/2.
(f) Put
ek = 1
k
∑
d |k
µ(d )2k/d .
Show that if |z| < 1, then
log(1 − 2z) =
∞∑
k=1
ek log
(
1 − zk )
.
(g) Deduce that
a =
∞∏
k=1
ζ(2k)ek .
Note that the kth factor here differs from 1 by an amount that is
≪ 1/(k2k ). Hence the product converges very rapidly . Since ζ(2k)
can be calculated very accurately by the Euler–Maclaurin formula (see
Appendix B), the formula above permits the rapid calculation of the
constanta. | {
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2.1 Mean values 45
23. Let B1 (x ) = x − 1/2, as in Appendix B.
(a) Show that
∑
n≤x
1
n = log x + C0 − B1 ({x })/x + O (1/x 2 ).
(b) Write ∑
n≤x d (n) = x log x + (2C0 − 1)x + /Delta1 (x ). Show that
/Delta1 (x ) =− 2
∑
n≤√x
B1 ({x /n}) + O (1).
(c) Show that
∫X
0 /Delta1 (x ) dx ≪ X .
(d) Deduce that
∑
n≤X
d (n)( X − n) =
∫ X
0
(∑
n≤x
d (n)
)
dx
= 1
2 X 2 log X +
(
C0 − 3
4
)
X 2 + O ( X ).
24. Let r (n) be the number of ordered pairs ( a,b) of integers for which a2 +
b2 = n.
(a) Show that
∑
n≤x
r (n) = 1 + 4[√x ] + 8
∑
1≤n≤√x /2
[√
x − n2
]
− 4
[√
x /2
]2
.
(b) Show that
∑
1≤n≤√x /2
√
x − n2 =
(π
8 + 1
2
)
x − B1
({√
x /2
})
− 1
2
√x + O (1).
(c) Write ∑
0≤n≤x r (n) = πx + R(x ). Show that
R(x ) =− 8
∑
1≤n≤√x /2
B1
({√
x − n2 })
+ O (1).
25. (a) Show that if ( a,q ) = 1, and β is real, then
q∑
n=1
B1
({a
q n + β
})
= B1 ({qβ}).
(b) Show that if A ≥ 1, | f ′(x ) − a/q |≤ A/q 2 for 1 ≤ x ≤ q , and ( a,q ) =
1, then
q∑
n=1
B1 ({ f (n)}) ≪ A. | {
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} |
46 The elementary theory of arithmetic functions
(c) Suppose that Q ≥ 1 is an integer, B ≥ 1, and that 1 /Q3 ≤± f ′′(x ) ≤
B/Q3 for 0 ≤ x ≤ N where the choice of sign is independent of
x . Show that numbers ar , qr , Nr can be determined, 0 ≤ r ≤ R for
some R, so that (i) ( ar ,qr ) = 1, (ii) qr ≤ Q, (iii) | f ′( Nr ) − ar /qr |≤
1/(qr Q), and (iv) N0 = 0, Nr = Nr −1 + qr −1 for 1 ≤ r ≤ R, N − Q ≤
N R ≤ N .
(d) Show that under the above hypotheses
N∑
n=0
B1 ({ f (n)}) ≪ B ( R + 1) + Q.
(e) Show that the number of s for which as /qs = ar /qr is ≪ Q2 /q 2 .
Let 1 ≤ q ≤ Q. Show that the number of r for which qr = q is
≪ ( Q/q )2 ( BNq /Q3 + 1).
(f) Conclude that under the hypotheses of (c),
N∑
n=0
B1 ({ f (n)}) ≪ B 2 NQ −1 log 2 Q + BQ 2 .
26. Show that if U ≤ √x , then
∑
U <n≤2U
B1 ({x /n}) ≪ x 1/3 log x .
Let /Delta1 (x ) be as in Exercise 23(b). Show that /Delta1 (x ) ≪ x 1/3 (log x )2 .
27. Let R(x ) be as in Exercise 24(c). Show that R(x ) ≪ x 1/3 log x .
2.2 The prime number estimates of Chebyshev and
of Mertens
Because of the irregular spacing of the prime numbers, it seems hopeless to
give a useful exact formula for thenth prime. As a compromise we estimate the
nth prime, or equivalently , estimate the number π(x ) of primes not exceeding x .
Similarly we put ϑ(x ) = ∑
p≤x log p, and ψ(x ) = ∑
n≤x /Lambda1 (n). As we shall see,
these three summatory functions are closely related. W e estimate ψ(x ) first.
Theorem 2.4 (Chebyshev) Fo r x ≥ 2, ψ(x ) ≍ x.
The proof we give below establishes only that there is an x0 such that
ψ(x ) ≍ x uniformly for x ≥ x0 . However, both ψ(x ) and x are bounded away
from 0 and from ∞ in the interval [2 ,x0 ], and hence the implicit constants can
be adjusted so that ψ(x ) ≍ x uniformly for x ≥ 2. In subsequent situations of | {
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2.2 Estimates of Chebyshev and of Mertens 47
this sort, we shall assume without comment that the reader understands that it
suffices to prove the result for all sufficiently largex .
Proof By applying the M ¨ obius inversion formula to (1.22) we find that
/Lambda1 (n) =
∑
d |n
µ(d ) log n/d .
Thus by (2.7) it follows that
ψ(x ) =
∑
d ≤x
µ(d )T (x /d ) (2.10)
where T (x ) = ∑
n≤x log n. By the integral test we see that
∫ N
1
log ud u ≤ T ( N ) ≤
∫ N +1
1
log ud u
for any positive integer N . Since
∫
log xd x = x log x − x , it follows easily
that
T (x ) = x log x − x + O (log 2 x ) (2.11)
for x ≥ 1. Despite the precision of this estimate, we encounter difficulties when
we substitute this in (2.10), since we have no useful information concerning the
sums
∑
d ≤x
µ(d )
d ,
∑
d ≤x
µ(d ) log d
d ,
which arise in the main terms. T o avoid this problem we introduce an idea that
is fundamental to much of prime number theory , namely we replaceµ(d )b y
an arithmetic function ad that in some way forms a truncated approximation to
µ(d ). Suppose that D is a finite set of numbers, and that ad = 0 when d /∈ D.
Then by (2.11) we see that
∑
d ∈D
ad T (x /d ) = (x log x − x )
∑
d ∈D
ad /d − x
∑
d ∈D
ad log d
d + O (log 2 x ).
(2.12)
Here the implicit constant depends on the choice of ad , which we shall consider
to be fixed. Since we want the above to approximate the relation (2.10), and
since we are hoping thatψ(x ) ≍ x , we restrict our attention to ad that satisfy
the condition
∑
d ∈D
ad
d = 0, (2.13) | {
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48 The elementary theory of arithmetic functions
and hope that
−
∑
d ∈D
ad log d
d is near 1 . (2.14)
By the definition of T (x ) we see that the left-hand side of (2.12) is
∑
dn ≤x
ad log n =
∑
dn ≤x
ad
∑
k|n
/Lambda1 (k) =
∑
dkm ≤x
ad /Lambda1 (k)
(2.15)
=
∑
k≤x
/Lambda1 (k) E (x /k)
where E ( y) = ∑
dm ≤y ad = ∑
d ad [ y/d ]. The expression above will be near
ψ(x )i f E ( y) is near 1. If y ≥ 1 then
∑
d
µ(d )[ y/d ] =
∑
d
µ(d )
∑
k≤y/d
1 =
∑
dk ≤y
µ(d ) =
∑
n≤y
∑
d |n
µ(d ) = 1,
in view of (1.20). Thus E ( y) will be near 1 for y not too large if ad is near µ(d )
for small d . Moreover, by (2.13) we see that E ( y) =− ∑
d ∈D ad {y/d }, so that
E ( y) is periodic with period dividing lcm d ∈D d . Hence for a given choice of
the ad , the behaviour of E ( y) can be determined by a finite calculation.
The simplest realization of this approach involves taking a1 = 1, a2 =− 2,
ad = 0 for d > 2. Then (2.13) holds, the expression (2.14) is log 2, E ( y) has
period 2 and E ( y) = 0 for 0 ≤ y < 1, E ( y) = 1 for 1 ≤ y < 2. Hence for this
choice of the ad the sum in (2.15) satisfies the inequalities
ψ(x ) − ψ(x /2) =
∑
x /2<k≤x
/Lambda1 (k) ≤
∑
k≤x
/Lambda1 (k) E (x /k) ≤
∑
k≤x
/Lambda1 (k) = ψ(x ).
Thus ψ(x ) ≥ (log 2) x + O (log x ), which is a lower bound of the desired shape.
In addition,
ψ(x ) − ψ(x /2) ≤ (log 2) x + O (log x ).
On replacing x by x /2r and summing over r we deduce that
ψ(x ) ≤ 2(log 2) x + O ((log x )2 ),
so the proof is complete. □
Chebyshev obtained better constants than above, by taking a1 = a30 = 1,
a2 = a3 = a5 =− 1, ad = 0 otherwise. Then (2.13) holds, the expression (2.14)
is 0 .92129 ... , E ( y) = 1 for 1 ≤ y < 6, and 0 ≤ E ( y) ≤ 1 for all y, with the
result that
ψ(x ) ≥ (0.9212)x + O (log x ) | {
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2.2 Estimates of Chebyshev and of Mertens 49
and
ψ(x ) ≤ (1.1056)x + O ((log x )2 ).
By computing the implicit constants one can use this method to determine a
constantx0 such that ψ(2x ) − ψ(x ) > x /2 for all x > x0 . Since the contribution
of the proper prime powers is small, it follows that there is at least one prime
in the interval (x , 2x ], when x > x0 . After separate consideration of x ≤ x0 ,
one obtains Bertrand’s postulate: For each real number x > 1, there is a prime
number in the interval ( x ,2x ).
Chebyshev said it, but I’ll say it again:
There’s always a prime between n and2n.
N. J. Fine
Corollary 2.5 Fo r x ≥ 2,
ϑ(x ) = ψ(x ) + O
(
x 1/2 )
and
π(x ) = ψ(x )
log x + O
( x
(log x )2
)
.
Proof Clearly
ψ(x ) =
∑
pk ≤x
log p =
∞∑
k=1
ϑ
(
x 1/k )
.
But ϑ( y) ≤ ψ( y) ≪ y, so that
ψ(x ) − ϑ(x ) =
∑
k≥2
ϑ(x 1/k ) ≪ x 1/2 + x 1/3 log x ≪ x 1/2 .
As for π(x ), we note that
π(x ) =
∫ x
2−
(log u)−1 d ϑ(u) = ϑ(x )
log x +
∫ x
2
ϑ(u)
u(log u)2 du .
This last integral is
≪
∫ x
2
(log u)−2 du ≪ x (log x )−2 ,
so we have the stated result. □
Corollary 2.6 Fo r x ≥ 2, ϑ(x ) ≍ x and π(x ) ≍ x /log x.
In Chapters 6 and 8 we shall give several proofs of the Prime Number
Theorem (PNT), which asserts that π(x ) ∼ x /log x . By Corollary 2.5 this is | {
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50 The elementary theory of arithmetic functions
equivalent to the estimates ϑ(x ) ∼ x , ψ(x ) ∼ x . By partial summation it is
easily seen that the PNT implies that
∑
p≤x
log p
p ∼ log x ,
and that
∑
p≤x
1
p ∼ log log x .
However, these assertions are weaker than PNT , as we can derive them from
Theorem 2.4.
Theorem 2.7Fo r x ≥ 2,
(a)
∑
n≤x
/Lambda1 (n)
n = log x + O (1),
(b)
∑
p≤x
log p
p = log x + O (1),
(c)
∫ x
1
ψ(u)u−2 du = log x + O (1),
(d)
∑
p≤x
1
p = log log x + b + O (1/log x ),
(e)
∏
p≤x
(
1 − 1
p
)−1
= eC0 log x + O (1)
where C 0 is Euler’s constant and
b = C0 −
∑
p
∞∑
k=2
1
kp k .
Proof T aking f (d ) = /Lambda1 (d ) in (2.1), we see from (2.3) that
T (x ) =
∑
n≤x
log n = x
∑
d ≤x
/Lambda1 (d )
d + O (ψ(x )) .
By Theorem 2.4 the error term is ≪ x . Thus (2.11) gives (a). The sum in (b)
differs from that in (a) by the amount
∑
pk ≤x
k≥2
log p
pk ≤
∑
p
log p
p( p − 1) ≪ 1.
T o derive (c) we note that the sum in (a) is
∫ x
2−
u−1 d ψ(u) = ψ(u)
u
⏐
⏐
⏐
x
2−
+
∫ x
2
ψ(u)u−2 du =
∫ x
2
ψ(u)u−2 du + O (1) | {
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2.2 Estimates of Chebyshev and of Mertens 51
by Theorem 2.4. W e now prove (d) without determining the value of the con-
stantb. W e express (b) in the form L (x ) = log x + R(x ) where R(x ) ≪ 1.
Then
∑
p≤x
1
p =
∫ x
2−
(log u)−1 dL (u) =
∫ x
2−
1
log u d log u +
∫ x
2−
dR (u)
log u
=
∫ x
2−
du
u log u +
[ R(u)
log u
⏐
⏐
⏐
⏐
x
2−
−
∫ x
2−
R(u) d (log u)−1
= log log x − log log 2 + 1 + R(x )
log x +
∫ x
2
R(u)
u(log u)2 du .
The penultimate term is ≪ 1/log x , and the integral is
∫∞
2 −
∫∞
x =∫∞
2 +O (1/log x ), so we have (d) with
b = 1 − log log 2 +
∫ ∞
2
R(u)
u(log u)2 du .
As for (e), we note that
∑
p≤x
log
(
1 − 1
p
)−1
=
∑
p≤x
1
p +
∑
p≤x
(
log
(
1 − 1
p
)−1
− 1
p
)
.
The second sum on the right is
∑
p
∞∑
k=2
1
kp k + O
(∑
p>x
p−2
)
and the error term here is ≪ ∑
n>x n−2 ≪ x −1 , so from (d) we have
∑
p≤x
log
(
1 − 1
p
)−1
= log log x + c + O (1/log x ) (2.16)
where c = b + ∑
p
∑
k≥2 (kp k )−1 . Since ez = 1 + O (|z|) for |z|≤ 1, on expo-
nentiating we deduce that
∏
p≤x
(
1 − 1
p
)−1
= ec log x + O (1).
T o complete the proof it suffices to show that c = C0 . T o this end we first note
that if p ≤ x and pk > x , then k ≥ (log x )/log p. Hence
∑
p≤x
pk >x
1
kp k ≪
∑
p≤x
pk >x
log p
(log x ) pk ≪
∑
p
log p
log x
∑
k≥2
p−k ≪ 1
log x
∑
p
log p
p2 ≪ 1
log x , | {
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52 The elementary theory of arithmetic functions
so that from (2.16) we have
∑
1<n≤x
/Lambda1 (n)
n log n = log log x + c + O (1/log x ).
By Corollary 1.15 this can be written
∑
1<n≤x
/Lambda1 (n)
n log n =
∑
n≤log x
1
n + (c − C0 ) + O (1/log 2 x ).
Since this is trivial when 1 ≤ x < 2, the above holds for all x ≥ 1. W e
express this briefly as T1 = T2 + T3 + T4 , and estimate the quantities Ii =
δ
∫∞
1 x −1−δTi (x ) dx . On comparing the results as δ → 0+ we shall deduce
that c = C0 . By Theorem 1.3, Corollary 1.11, and Corollary 1.13 we see that
I1 = log ζ(1 + δ) = log 1
δ + O (δ)
as δ → 0+. Secondly ,
I2 = δ
∞∑
n=1
1
n
∫ ∞
en
x −1−δ dx =
∞∑
n=1
1
n e−δn = log(1 − e−δ)−1
= log(δ + O (δ2 ))−1 = log 1 /δ + O (δ).
Thirdly ,
I3 = c − C0 ,
and finally
I4 ≪ δ
∫ ∞
1
x −1−δ dx
log 2 x ≪ δ + δ
∫ e1/δ
2
dx
x log x + δ2
∫ ∞
e1/δ
x −1−δ dx ≪ δlog 1 /δ.
Since the main terms cancel, on letting δ → 0+ we see that c = C0 .
□
Corollary 2.8 W e have
lim sup
x →∞
π(x )
x /log x ≥ 1
and
lim inf
x →∞
π(x )
x /log x ≤ 1.
Proof By Corollary 2.5 it suffices to show that lim sup ψ(u)/u ≥ 1, and that
lim inf ψ(u)/u ≤ 1. Suppose that lim sup ψ(u)/u = a, and suppose that ε> 0. | {
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2.2 Estimates of Chebyshev and of Mertens 53
Then there is an x0 such that ψ(x ) ≤ (a + ε)x for all x ≥ x0 , and hence
∫ x
1
ψ(u)u−2 du ≤
∫ x0
1
ψ(u)u−2 du +(a + ε)
∫ x
x0
u−1 du ≤ (a + ε) log x + Oε(1).
Since this holds for arbitrary ε> 0, it follows that
∫x
1 ψ(u)u−2 du ≤ (a +
o(1)) log x . Thus by Theorem 2.7(c) we have a ≥ 1. Similarly lim inf ψ(u)/u
≤ 1. □
2.2.1 Exercises
1. (a) Let dn = [1,2,... , n]. Show that dn = eψ(n) .
(b) Let P ∈ Z[x ], deg P ≤ n. Put I = I ( P ) =
∫1
0 P (x ) dx . Show that
Id n+1 ∈ Z, and hence that dn+1 ≥ 1/|I | if I ̸=0.
(c) Show that there is a polynomial P as above so that Id n+1 = 1.
(d) V erify that max 0≤x ≤1 |x 2 (1 − x )2 (2x − 1)|= 5−5/2 .
(e) For P (x ) =
(
x 2 (1 − x )2 (2x − 1)
)2n
, verify that 0 < I < 5−5n .
(f) Show that ψ(10n + 1) ≥ ( 1
2 log 5) · 10n.
2. Let A be the set of integers composed entirely of primes p ≤ A1 , and
let B be the set of integers composed entirely of primes p > A1 . Then n
is uniquely of the form n = ab , a ∈ A, b ∈ B. Let δ( A1 , A2 ) denote the
density of those n such that a ≤ A2 .
(a) Give a formula for δ( A1 , A2 ).
(b) Show that δ( A1 , A2 ) ≫ (log A2 )/log A1 for 2 ≤ A2 ≤ A1 .
3. Let an = 1 + cos log n, and note that an ≥ 0 for all n.
(a) Show that
∞∑
n=1
an n−s = ζ(s) + 1
2 ζ(s + i ) + 1
2 ζ(s − i )
for σ> 1.
(b) By Corollary 1.15, or otherwise, show that
∑
n≤x
an
n = log x + O (1).
(c) By integrating by parts as in the proof of Theorem 1.12, show that
∑
n≤x
an =
(
1 + x i
2(1 + i ) + x −i
2(1 − i )
)
x + O (log x ).
(d) Deduce that
lim inf
x →∞
1
x
∑
n≤x
an = 1 − 1√
2
, lim sup
x →∞
1
x
∑
n≤x
an = 1 + 1√
2
. | {
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54 The elementary theory of arithmetic functions
Thus for the coefficients an we have an analogue of Mertens’ esti-
mate of Theorem 2.7(b), but not an analogue of the Prime Number
Theorem.
4. (Golomb 1992) Let dx denote the least common multiple of the positive
integers not exceeding x . Show that
(2n
n
)
=
∞∏
k=1
d (−1)k−1
2n/k .
5. (Chebyshev 1850) From Corollaries 2.5 and 2.8 we see that if there is a
number a such that ψ(x ) = (a + o(1))x as x →∞ , then we must have
a = 1. W e now take this a step further.
(a) Suppose that there is a number a such that
ψ(x ) = x + (a + o(1))x /log x (2.17)
as x →∞ . Deduce that
∫ x
2
ψ(u)
u2 du = log x + (a + o(1)) log log x
as x →∞ .
(b) By comparing the above with Theorem 2.7(c), deduce that if (2.17)
holds, then necessarily a = 0.
(c) Suppose that there is a constant A such that
π(x ) = x
log x − A + o
( x
(log x )2
)
(2.18)
as x →∞ . By writing ϑ(x ) =
∫x
2− log ud π(u), integrating by parts,
and estimating the expressions that arise, show that if (2.18) holds,
then
ψ(x ) = x + ( A − 1 + o(1))x /log x
as x →∞ .
(d) Deduce that if (2.18) holds, then A = 1.
2.3 Applications to arithmetic functions
The results above are useful in determining the extreme values of familiar
arithmetic functions. W e consider three instances. | {
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2.3 Applications to arithmetic functions 55
Theorem 2.9 F or all n ≥ 3,
ϕ(n) ≥ n
log log n
(
e−C0 + O (1/log log n)
)
,
and there are infinitely many n for which the above relation holds with equality.
ProofLet R be the set of those n for which ϕ(n)/n <ϕ (m)/m for all m < n.
W e first prove the inequality for these ‘record-breaking’ n ∈ R. Suppose that
ω(n) = k, and let n∗ be the product of the first k primes. If n ̸=n∗ then n∗ < n
and ϕ(n∗)/n∗ <ϕ (n)/n. Hence R is the set of n of the form
n =
∏
p≤y
p. (2.19)
T aking logarithms, we see that log n = ϑ( y) ≍ y by Corollary 2.6. On taking
logarithms a second time, it follows that log log n = log y + O (1). Thus by
Mertens’ formula (Theorem 2.7(e)) we see that
ϕ(n)
n =
∏
p≤y
(
1 − 1
p
)
= e−C0
log y
(
1 + O (1/log y)
)
,
which gives the desired result for n ∈ R.I f n /∈ R then there is an m < n such
that m ∈ R, ϕ(m)/m <ϕ (n)/n. Hence
ϕ(n)
n > ϕ(m)
m = 1
log log m
(
e−C0 + O
( 1
log log m
))
≥ 1
log log n
(
e−C0 + O
( 1
log log n
))
.
W e note that equality holds for n of the type (2.19), so the proof is complete. □
Theorem 2.10 F or all n ≥ 3,
1 ≤ ω(n) ≤ log n
log log n (1 + O (1/log log n)) .
Proof As in the preceding proof we see that record-breaking values of ω(n)
occur when n is of the form (2.19), and that it suffices to prove the bound for
these n. As in the preceding proof, for n given by (2.19) we have ϑ( y) = log n
and log y = log log n + O (1). This gives the result, and we note that the bound
is sharp for these n. □
W e now consider the maximum order of d (n). From the pairing d ↔ n/d
of divisors, and the fact that at least one of these is ≤ √n, it is immediate that
d (n) ≤ 2√n. On the other hand, if n is square-free then d (n) = 2ω(n) , which | {
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56 The elementary theory of arithmetic functions
can be large, but not nearly as large as √n. Indeed, for each ε> 0 there is a
constant C (ε) such that
d (n) ≤ C (ε)nε (2.20)
for all n ≥ 1. T o see this we express n in terms of its canonical factorization,
n = ∏
p pa , so that
d (n)
nε =
∏
p
a + 1
paε =
∏
p
f p (a),
say . Let αp be an integral value of a for which f p (a) is maximized. From the
inequalities f p (αp ) ≥ f p (αp ± 1) we see that
( pε − 1)−1 − 1 ≤ αp ≤ ( pε − 1)−1 ,
so that we may take αp = [( pε − 1)−1 ]. Hence (2.20) holds with
C (ε) =
∏
p
f p (αp ).
This constant is best possible, since equality holds when n = ∏
p pαp .B y
analysing the rate at which C (ε) grows as ε → 0+, we derive
Theorem 2.11 F or all n ≥ 3
log d (n) ≤ log n
log log n (log 2 + O (1/log log n)) .
W e note that this bound is sharp for n of the form in (2.19).
Proof It suffices to show that there is an absolute constant K such that
C (ε) ≤ exp
(
K ε2 21/ε)
, (2.21)
since the stated bound then follows by taking ε = (log 2) /log log n. W e observe
that αp = 0i f p > 21/ε, that αp = 1i f( 3 /2)1/ε < p ≤ 21/ε, and that αp ≪ 1/ε
when p ≤ (3/2)1/ε. Hence
log C (ε) ≪
∑
p≤21/ε
log(2/pε) +
∑
p≤(3/2)1/ε
log(1/ε).
Here the second sum is π
(
(3/2)1/ε)
log 1 /ε ≪ ε2 21/ε. The first sum is
(log 2) π(21/ε) − εϑ(21/ε), and by Corollary 2.5 this is ≪ ε2 21/ε. Thus we have
(2.21), and the proof is complete. □
It is very instructive to consider our various results from the perspective of
elementary probability theory . Let d be a fixed integer. Then the set of n that
are divisible by d has asymptotic density 1 /d , and we might say , loosely , that | {
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2.3 Applications to arithmetic functions 57
the ‘probability’ that d |n when n is ‘randomly chosen’ is 1 /d .I f d1 and d2
are two fixed numbers then the ‘probability’ that d1 |n and d2 |n is 1 /[d1 ,d2 ].
If ( d1 ,d2 ) = 1 then this ‘probability’ is 1 /(d1 d2 ), and we see that the ‘events’
d1 |n, d2 |n are ‘independent. ’ T o make this rigourous we consider the integers
1 ≤ n ≤ N , and assign probability 1 /N to each of the N numbers n. Then
P(d |n) = [ N /d ]/N = 1
d − 1
N {N /d }.
This is 1 /d if d |N ; otherwise it is close to 1/d if d is small compared to N .
Similarly the events d1 |n, d2 |n are not independent in general, but are nearly
independent if N /(d1 d2 ) is large. The probabilistic heuristic, in which inde-
pendence is assumed, provides a useful means of constructing conjectures.
Many of our investigations can be considered to be directed toward determin-
ing whether the cumulative effect of the error terms{N /d }/N have a discernible
effect.
As an example of the probabilistic approach, we note that n is square-free
if and only if none of the numbers 2 2 ,32 ,52 ,... , p2 ,... divide n. The ‘prob-
ability’ that p2 ∤ n is approximately 1 − 1/p2 . Since these events are nearly
independent, we predict that the probability that a random integer n ∈ [1,N ]i s
square-free is approximately ∏
p≤N (1 − 1/p2 ). This was confirmed in Theorem
2.2. On the other hand, the sieve of Eratosthenes asserts that
∑
n≤N
(n,P )=1
1 = π( N ) − π
(√
N
)
+ 1
where P = ∏
p≤
√
N p. For a random n ∈ [1,N ] we expect that the probability
that ( n,P ) = 1 should be approximately
ϕ( P )
P =
∏
p≤
√
N
(
1 − 1
p
)
∼ 2e−C0
log N
by Mertens’ formula (Theorem 2.7(e)). This would suggest that perhaps
π(x ) ∼ 2e−C0 x
log x .
However, since 2 e−C0 = 1.1229189 ... , this conflicts with the Prime Number
Theorem, and also with Corollary 2.8. Thus the probabilistic model is mislead-
ing in this case.
Suppose now that X p (n) is the arithmetic function
X p (n) =
{1i f p|n,
0 otherwise , | {
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58 The elementary theory of arithmetic functions
so that ω(n) = ∑
p X p (n). If we were to treat the X p as though they
were independent random variables then we would have E( X p ) = 1/p,
V ar(X p ) = (1 − 1/p)/p. Hence we expect that the average of ω(n) should be
approximately
E
(∑
p≤n
X p
)
=
∑
p≤n
E( X p ) =
∑
p≤n
1
p = log log n + O (1),
and that its variance is approximately
Va r
(∑
p≤n
X p
)
=
∑
p≤n
V ar(X p ) =
∑
p≤n
(
1 − 1
p
)1
p = log log n + O (1).
The first of these is easily confirmed, since by (2.3) we have
∑
n≤x
ω(n) = x
∑
p≤x
1
p + O (π(x )) .
By Mertens’ formula (Theorem 2.7(d)) and Chebyshev’s bound (Corollary 2.6)
this is
= x log log x + bx + O (x /log x ). (2.22)
As for the variance, we have
Theorem 2.12(Tur´ an)Fo r x ≥ 3,
∑
n≤x
(ω(n) − log log x )2 ≪ x log log x (2.23)
and
∑
1<n≤x
(ω(n) − log log n)2 ≪ x log log x . (2.24)
These estimates also hold with ω(n) replaced by /Omega1 (n).
Let E be the set of ‘exceptional’ n for which
|ω(n) − log log n| > (log log n)3/4 .
By Theorem 2.12 we see that
∑
n∈E
x <n≤2x
1 ≤ (log log x )−3/2 ∑
n≤2x
(ω(n) − log log n)2 ≪ x
(log log x )1/2 = o(x ),
so we have | {
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2.3 Applications to arithmetic functions 59
Corollary 2.13 (Hardy–Ramanujan) F or almost all n, ω(n) ∼ /Omega1 (n) ∼
log log n.
Note that in analytic number theory we say ‘almost all’ when the excep-
tional set has asymptotic density 0; this conflicts with the usage in some
parts of algebra, where the term means that there are at most finitely many
exceptions.
Proof of Theorem 2.12T o prove (2.23) we first multiply out the square on the
left, and write the sum as
/Sigma1 2 − 2(log log x )/Sigma1 1 + [x ](log log x )2 . (2.25)
W e have already determined the size of /Sigma1 1 in (2.22). The new sum is
/Sigma1 2 =
∑
n≤x
ω(n)2 =
∑
n≤x
(∑
p1 |n
1
)(∑
p2 |n
1
)
=
∑
p1 ≤x
p2 ≤x
∑
n≤x
pi |n
1.
The terms for which p1 = p2 contribute
∑
p≤x
[x /p] = x
∑
p≤x
1
p + O (π(x )) = x log log x + O (x ).
The terms p1 ̸=p2 contribute
∑
p1 ̸=p2
[ x
p1 p2
]
≤ x
∑
p1 p2 ≤x
p1 ̸=p2
1
p1 p2
≤ x
(∑
p≤x
1
p
)2
= x (log log x )2 + O (x log log x )
(2.26)
by Mertens’ formula (Theorem 2.7(d)). Thus
/Sigma1 2 ≤ x (log log x )2 + O (x log log x ).
The estimate (2.23) now follows by inserting this and (2.22) in (2.25).
W e derive (2.24) from (2.23) by applying the triangle inequality
⏐
⏐∥ x∥−
∥ y∥
⏐
⏐≤∥ x − y∥ for vectors. This gives
⏐
⏐
⏐
⏐
(∑
1<n≤x
(ω(n) − log log n)2
)1/2
−
(∑
1<n≤x
(ω(n) − log log x )2
)1/2 ⏐
⏐
⏐
⏐
≤
(∑
1<n≤x
(log log x − log log n)2
)1/2
. | {
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60 The elementary theory of arithmetic functions
By the integral test the sum on the right is
=
∫ x
e
(log log x − log log u)2 du + O ((log log x )2 ).
By integrating by parts twice we find that this integral is
−e(log log x )2 −2e log log x +2
∫ x
2
1 + log log x −log log u
(log u)2 du ≪ x
(log x )2 .
Thus
(∑
1<n≤x
(ω(n)−log log n)2
)1/2
=
(∑
n≤x
(ω(n) − log log x )2
)1/2
+ O
(
x 1/2 /log x
)
,
and (2.24) follows by squaring both sides and applying (2.23). W e omit the
similar argument for/Omega1 (n). □
Since 2 ω(n) ≤ d (n) ≤ 2/Omega1 (n) for all n, Corollary 2.13 carries an interesting
piece of information for d (n):
d (n) = (log n)(log 2 +o(1))
for almost all n. Since this is smaller than the average size of d (n), we see that
the average is determined not by the usual size of d (n) but by a sparse set of n for
which d (n) is disproportionately large. Since the first moment (i.e., average) of
d (n) is inflated by the ‘tail’ in its distribution, it is not surprising that this effect
is more pronounced for the higher moments. As was originally suggested by
Ramanujan, it can be shown that for any fixed real numberκ there is a positive
constant c(κ) such that
∑
n≤x
d (n)κ ∼ c(κ)x (log x )2κ−1 (2.27)
as x →∞ .
In order to handle the error terms that arise in our arguments we are frequently
led to estimate the mean value of multiplicative functions. In most such cases
the method of the hyperbola or the simpler identity (2.3) will suffice, but the
labour involved quickly becomes tiresome. It will therefore be convenient to
have the following result on record, as it is very readily applied.
Theorem 2.14Let f be a non-negative multiplicative function. Suppose that
A is a constant such that
∑
p≤x
f ( p) log p ≤ Ax (2.28) | {
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2.3 Applications to arithmetic functions 61
for all x ≥ 1, and that
∑
pk
k≥2
f ( pk )k log p
pk ≤ A. (2.29)
Then for x ≥ 2,
∑
n≤x
f (n) ≪ ( A + 1) x
log x
∑
n≤x
f (n)
n .
W e note that this is sharper than the trivial estimate
∑
n≤x
f (n) ≤ x
∑
n≤x
f (n)/n (2.30)
that holds whenever f ≥ 0.
If f ≥ 0 and f is multiplicative, then
∑
n≤x
f (n)
n ≤
∏
p≤x
(
1 + f ( p)
p + f ( p2 )
p2 +···
)
.
On combining this with Theorem 2.14 we obtain
Corollary 2.15Under the above hypotheses
∑
n≤x
f (n) ≪ ( A + 1) x
log x
∏
p≤x
(
1 + f ( p)
p + f ( p2 )
p2 +···
)
.
Suppose for example that f (n) = d (n)κ. W e write
∏
p≤x
(
1 + 2κ
p + 3κ
p2 +···
)
=
(∏
p≤x
(
1 − 1
p
)−2κ )(∏
p≤x
(
1 − 1
p
)2κ
×
(
1 + 2κ
p + 3κ
p2 +···
))
and observe that the second product tends to a finite limit as x →∞ , so that
by Mertens’ formula (Theorem 2.7(e)) we have
∑
n≤x
d (n)κ ≪ x (log x )2κ−1 (2.31)
for any fixed κ. Though weaker than (2.27), this is all that is needed in many
cases. W e can similarly show that for any fixed real κ,
∑
n≤x
(n
ϕ(n)
)κ
≪ x . (2.32) | {
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62 The elementary theory of arithmetic functions
Thus we see that ϕ(n)/n is not often very small.
Proof of Theorem 2.14 The desired bound is obtained by adding the two
estimates
∑
n≤x
f (n) log x
n ≪ x
∑
n≤x
f (n)
n , (2.33)
∑
n≤x
f (n) log n ≪ Ax
∑
n≤x
f (n)
n . (2.34)
The first of these is immediate, since f ≥ 0 and log x /n ≪ x /n uniformly for
1 ≤ n ≤ x . Since log n = ∑
d |n /Lambda1 (d ), the second sum is
∑
d ≤x
/Lambda1 (d )
∑
m≤x /d
f (md ).
Writing d = pi , m = p j r where p ∤ r , we see that this is
∑
p,i ≥1,j ≥0
pi + j ≤x
(log p) f ( pi + j )
∑
r ≤x /pi + j
p∤r
f (r ) =
∑
p,k
pk ≤x
k(log p) f ( pk )
∑
r ≤x /pk
p∤r
f (r ).
Here we have put i + j = k. W e now drop the condition p ∤ r on the right-
hand side, and consider first the contribution of the proper prime powers (i.e.,
k≥ 2). By (2.30) with x replaced by x /p we see that the terms for which k ≥ 2
contribute
≪ x
∑
p,k≥2
(log pk ) f ( pk ) p−k ∑
r ≤x /pk
f (r )/r ≤ Ax
∑
n≤x
f (n)/n
by (2.29). It remains to bound
∑
p≤x
(log p) f ( p)
∑
r ≤x /p
f (r ) =
∑
r ≤x
f (r )
∑
p≤x /r
f ( p) log p.
By (2.28) this is ≤ Ax ∑
r ≤x f (r )/r , so we have (2.34) and the proof is
complete. □
In the above proof we made no use of prime number estimates, but as we
have seen the estimates of Chebyshev are useful in verifying the hypotheses
and Mertens’ formula is helpful in estimating the sum∑
n≤x f (n)/n.
2.3.1 Exercises
1. Let σ(n) = ∑
d |n d .
(a) Show that σ(n)ϕ(n) ≤ n2 for all n ≥ 1.
(b) Deduce that n + 1 ≤ σ(n) ≤ eC0 n
(
log log n + O (1)
)
for all n ≥ 3. | {
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2.3 Applications to arithmetic functions 63
2. Show that d (n) ≤
√
3n with equality if and only if n = 12.
3. Let f (n) = ∏
p|n (1 + p−1/2 ).
(a) Show that there is a constant a such that if n ≥ 3, then
f (n) < exp
(
a(log n)1/2 (log log n)−1 )
.
(b) Show that ∑
n≤x f (n) = cx + O
(
x 1/2 )
where c = ∏
p (1 + p−3/2 ).
4. Let dk (n) be as in Exercise 2.1.18. Show that if k and κ are fixed, then
∑
n≤x
dk (n)κ ≪ x (log x )kκ−1 .
for x ≥ 2.
5. (Davenport 1932) Let
f (n) =−
∑
d |n
µ(d ) log d
d .
(a) By recalling Exercise 2.1.16(a), or otherwise, show that f (n) ≥ 0 for
all n.
(b) Show that f (n) ≪ log log n for n ≥ 3.
(c) Show that f (n) ∼ 1
4 log log n if n = ∏
y<p≤y2 p.
(d) Show that f (n) ≤
(1
4 + o(1)
)
log log n as n →∞ .
6. (cf. Bateman & Grosswald 1958) Let F be the set of ‘power-full’ numbers
where n is power-full if p|n ⇒ p2 |n.
(a) Show that
∑
n∈F
n−s = ζ(2s)ζ(3s)
ζ(6s)
for σ> 1/2.
(b) Show that
∑
a,b,c
a2 b3 c6 =n
µ(c) =
{1i f n ∈ F,
0 otherwise .
(c) Show that
∑
a2 b3 ≤x
1 = ζ(3/2) y1/2 + ζ(2/3) y1/3 + O
(
y1/5 )
.
(d) Show that
∑
n≤x
n∈F
1 = ζ(3/2)
ζ(3) x 1/2 + ζ(2/3)
ζ(2) x 1/3 + O
(
x 1/5 )
. | {
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64 The elementary theory of arithmetic functions
7. (Bateman 1949) Let /Phi1 q (z) denote the q th cyclotomic polynomial,
/Phi1 q (z) =
q∏
a=1
(a,q )=1
(z − e(a/q ))
where e(θ) = e2πi θ.
(a) Show that
∏
d |q
/Phi1 d (z) = zq − 1.
(b) Show that
/Phi1 q (z) =
∏
d |q
(zd − 1)µ(q /d ) .
(c) If P (z) = ∑ pn zn and Q(z) = ∑ qn zn are polynomials with real coeffi-
cients, then we say that P ≼ Q if | pn |≤ qn for all non-negative integers
n. Show that if P1 ≼ Q1 and P2 ≼ Q2 , then P1 + P2 ≼ Q1 + Q2 and
P1 P2 ≼ Q1 Q2 .
(d) Show that /Phi1 q (z) ≼ Qq (z) where
Qq (z) =
∏
d |q
(1 + zd + z2d +···+ zq −d ).
(e) Show that Qq (1) = q d (q )/2 .
(f) Show that for any ε> 0 there is a q0 (ε) such that if q > q0 (ε), then all
coefficients of /Phi1 q have absolute value not exceeding
exp
(
q (log 2 +ε)/log log q )
.
8. (Tur´ an 1934) (a) Show that the first sum in (2.26) is
= x
∑
p1 p2 ≤x
1
p1 p2
+ O (x ).
(b) Explain why the sum above is
(∑
p≤x
1
p
)2
− 2
∑
p1 ≤√x
1
p1
∑
x /p1 <p2 ≤x
1
p2
+
⎛
⎝∑
√
x <p≤x
1
p
⎞
⎠
2
. (2.35)
(c) Show that if y ≤ √
x , then
∑
x /y<p≤x
1
p = log log x − log log( x /y) + O (1/log x ).
(d) Show that the right-hand side above is ≍ (log y)/log x . | {
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2.4 The distribution of /Omega1 (n) − ω(n)6 5
(e) Deduce that the second and third terms in (2.35) are ≪ 1.
(f) Conclude that
/Sigma1 2 = x (log log x )2 + (2b + 1) log log x + O (x )
where b is the constant in Theorem 2.7(d).
(g) Show that the left-hand side of (2.23) is = x log log x + O (x ).
(h) Show that the left-hand side of (2.24) is = x log log x + O (x ).
9. (cf. Pomerance 1977, Shan 1985) Note that ϕ(n)|(n − 1) when n is prime. An
old – and still unsolved – problem of D. H. Lehmer asks whether there exists
a composite integern such that ϕ(n)|(n − 1). Let S denote the (presumably
empty) set of such numbers.
(a) Show that ifn ∈ S, then n is square-free.
(b) Suppose that mp ∈ S. Show that m ≡ 1 (mod p − 1).
(c) Let p be given. Show that the number of m such that mp ≤ x and mp ∈ S
is ≪ x /p2 .
(d) Show that the number of n ∈ S, n ≤ x , such that n has a prime factor
> y is ≪ x /( y log y).
(e) Suppose that x /y < n ≤ x and that n is composed entirely of primes
p ≤ y. Show that ω(n) ≥ (log x )/(log y) − 1.
(f) By Exercise 4, or otherwise, show that the number of n ≤ x such that
ω(n) ≥ z is ≪ x (log x )2 /3z .
(g) Conclude that the number of n ≤ x such that n ∈ S is
≪ x /exp(√
log x ).
2.4 The distribution of /Omega1 (n) − ω(n)
In order to illustrate further the use of elementary techniques we now discuss
an elegant result of R´enyi, which asserts that the set of numbers n such that
/Omega1 (n) − ω(n) = k has density dk , where the dk are the power series coefficients
of the meromorphic function
F (z) =
∞∑
k=0
dk zk =
∏
p
(
1 − 1
p
)(
1 + 1
p − z
)
. (2.36)
By examining this product we see that F has simple poles at the points z = p
( p ̸=3), and simple zeros at the points z = p + 1( p ̸=2), so that the power
series converges for |z| < 2. W e let Nk (x ) denote the number of n ≤ x for
which /Omega1 (n) − ω(n) = k; our object is to show that Nk (x ) ∼ dk x . If this holds
for each k then we can deduce that ∑ dk ≤ 1. By taking z = 1 in (2.36) we see
that ∑ dk = 1, which gives us hope that the asymptotic relation may be fairly | {
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66 The elementary theory of arithmetic functions
uniform in k. This is indeed the case, as we see from the following quantitative
form of R´ enyi’s theorem.
Theorem 2.16F or any non-negative integer k , and any x ≥ 2,
Nk (x ) = dk x + O
((3
4
)k
x 1/2 (log x )4/3 )
.
In preparation for the proof of this result we first establish a subsidiary
estimate.
Lemma 2.17F or any y ≥ 0 and any natural number f ,
∑
n≤y
(n,f )=1
µ(n)2 = 6
π2
(∏
p| f
(
1 + 1
p
)−1 )
y + O
(
y1/2 ∏
p| f
(
1 − p−1/2 )−1
)
.
Proof Let D ={ d : p|d ⇒ p| f }. By considering the Dirichlet series identity
∞∑
n=1
(n,f )=1
µ(n)2 n−s =
∏
p∤ f
(1 + p−s ) = ζ(s)
ζ(2s)
∏
p| f
(1 + p−s )−1 = ζ(s)
ζ(2s)
∑
d ∈D
λ(d )d −s ,
or by elementary considerations, we see that the characteristic function of the
set of those square-freen such that ( n, f ) = 1 may be written
∑
dm =n
d ∈D
λ(d )µ(m)2 .
Hence the sum in question is
∑
d ∈D
λ(d )
∑
m≤y/d
µ(m)2 =
∑
d ∈D
λ(d )
(6
π2 · y
d + O
(
y1/2 d −1/2 ))
by Theorem 2.2. But ∑
d ∈D λ(d )/d = ∏
p| f (1 + 1/p)−1 and ∑
d ∈D d −1/2 =∏
p| f (1 − p−1/2 )−1 , so that the proof is complete. □
Proof of Theorem 2.16 Let Q denote the set of square-free numbers and F
denote the set of ‘power-full’ numbers (i.e., those f such that p| f ⇒ p2 | f ).
Every number is uniquely expressible in the form n = qf , q ∈ Q, f ∈ F,
(q , f ) = 1. Hence
Nk =
∑
f ≤x
f ∈F
/Omega1 ( f )−ω( f )=k
∑
q ≤x /f
q ∈Q
(q ,f )=1
1. | {
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2.4 The distribution of /Omega1 (n) − ω(n)6 7
By Lemma 2.17 this is
6
π2 x
∑
f ≤x
f ∈F
/Omega1 ( f )−ω( f )=k
1
f
∏
p| f
(1 + p−1 )−1 + O
⎛
⎜
⎜
⎜
⎜
⎝
x 1/2 ∑
f ≤x
f ∈F
/Omega1 ( f )−ω( f )=k
f −1/2 ∏
p| f
(
1 − p−1/2 )−1
⎞
⎟
⎟
⎟
⎟
⎠
.
In order to appreciate the nature of these sums it is helpful to observe that each
member ofF is uniquely of the form a2 b3 with b square-free, so that there are
≍ x 1/2 members of F not exceeding x . Suppose that z ≥ 1. Then the sum in
the error term is
≤ z−k ∑
f ≤x
f ∈F
z/Omega1 ( f )−ω( f ) f −1/2 ∏
p| f
(
1 − p−1/2 )−1
.
Since /Omega1 ( f ) − ω( f ) is an additive function, it follows that z/Omega1 ( f )−ω( f ) is a mul-
tiplicative function. Hence the above is
≤ z−k ∏
p≤x
(
1 +
(
1 − p−1/2
)−1 (z
p + z2
p3/2 + z3
p2 +···
))
.
When p = 2 the sum converges only for z <
√
2. Hence we take z = 4/3, and
then the product is
≤
∏
p≤x
(
1 + 4
3 p + C
p3/2
)
≪ (log x )4/3
by Mertens’ formula. Thus
∑
f ≤x
f ∈F
/Omega1 ( f )−ω( f )=k
f −1/2 ∏
p| f
(
1 − p−1/2 )−1
≪
(3
4
)k
(log x )4/3
which suffices for the error term.
W e now consider the effect of dropping the condition f ≤ x in the main
term. Since
∑
U <f ≤2U
f ∈F
/Omega1 ( f )−ω( f )=k
1
f
∏
p| f
(
1 + 1
p
)−1
≤ U −1/2 ∑
U <f ≤2U
f ∈F
/Omega1 ( f )−ω( f )=k
f −1/2 ∏
p| f
(
1 − p−1/2 )−1
≪ U −1/2
(3
4
)k
(log 2 U )4/3 ,
on taking U = x 2r and summing over r ≥ 0 we see that
∑
f ≤x
f ∈F
/Omega1 ( f )−ω( f )=k
1
f
∏
p| f
(
1 + 1
p
)−1
≪ x −1/2
(3
4
)k
(log x )4/3 . | {
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68 The elementary theory of arithmetic functions
Hence we have the stated result with
dk = 6
π2
∑
f ∈F
/Omega1 ( f )−ω( f )=k
1
f
∏
p| f
(
1 + 1
p
)−1
.
T o see that (2.36) holds, it suffices to multiply this by zk and sum over k. □
2.4.1 Exercise
1. Let dk be as in (2.36). Show that
dk = c2−k + O (5−k )
where
c = 1
4
∏
p>2
(
1 − 1
( p − 1)2
)−1
.
2.5 Notes
Section 2.1. Mertens (1874 a) showed that ∑
n≤x ϕ(n) = 3x 2 /π2 + O (x log x ).
This refines an earlier estimate of Dirichlet, and is equivalent to Theorem 2.1,
by partial summation. LetR(x ) denote the error term in Theorem 2.1. Chowla
(1932) showed that
∫ x
1
R(u)2 du ∼ x
2π2
as x →∞ , and W alfisz (1963, p. 144) showed that
R(x ) ≪ (log x )2/3 (log log x )4/3 .
In the opposite direction, Pillai & Chowla (1930) showed (cf. Exercise
7.3.6) thatR(x ) = /Omega1 (log log log x ). That the error term changes sign in-
finitely often was first proved by Erd ˝ os & Shapiro (1951), who showed that
R(x ) = /Omega1 ±(log log log log x ). More recently , Montgomery (1987) showed that
R(x ) = /Omega1 ±(√
log log x ). It may be speculated that R(x ) ≪ log log x and that
R(x ) = /Omega1 ±(log log x ).
Theorem 2.2 is due to Gegenbauer (1885).
Theorem 2.3 is due to Dirichlet (1849). The problem of improving the error
term in this theorem is known as the Dirichlet divisor problem . Let /Delta1 (x ) denote
the error term. V orono¨ ı (1903) showed that /Delta1 (x ) ≪ x 1/3 log x (see Exercises
2.1.23, 2.1.25, 2.1.26). van der Corput (1922) used estimates of exponential
sums to show that/Delta1 (x ) ≪ x 33/100+ε. This exponent has since been reduced | {
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2.5 Notes 69
by van der Corput (1928), Chih (1950), Richert (1953), Kolesnik (1969, 1973,
1982, 1985), Iwaniec & Mozzochi (1988), and by Huxley (1993), who showed
that/Delta1 (x ) ≪ x 23/73+ε. In the opposite direction, Hardy (1916) showed that
/Delta1 (x ) = /Omega1 ±(x 1/4 ). Soundararajan (2003) showed that
/Delta1 (x ) = /Omega1
(
x 1/4 (log x )1/4 (log log x )b (log log log x )−5/8 )
with b = 3
4 (24/3 − 1), and it is plausible that the first three exponents above are
optimal.
The result of Exercise 2.1.12 generalizes to Rn : A lattice point
(a1 ,a,... , an ∈ Zn ) is said to be primitive if gcd( a1 ,a2 ,... , an ) = 1. The
asymptotic density of primitive lattice points is easily shown to be 1 /ζ(n).
In addition, Cai & Bach (2003) have shown that the density of lattice points
a ∈ Zn such that gcd( ai ,a j ) = 1 for all pairs with 1 ≤ i < j ≤ n is
∏
p
((
1 − 1
p
)n
+ n
p
(
1 − 1
p
)n−1 )
.
Section 2.2. Chebyshev (1848) used the asymptotics of log ζ(σ)a s σ → 1+
to obtain Corollary 2.8. In his second paper on prime numbers, Chebyshev
(1850) introduced the notationsϑ(x ), ψ(x ), T (x ), and proved Theorem 2.4,
Corollaries 2.5, 2.6, Theorem 2.7(a), and the results of Exercise 2.2.5. Sylvester
(1881) devised a more complicated choice of thead that gave better constants
than those of Chebyshev . Diamond & Erd ˝ os (1980) have shown that for any
ε>0 it is possible to choose numbers ad as in the proof of Theorem 2.4 to
show that (1 − ε)x <ψ (x ) < (1 + ε)x for all sufficiently large x . This does
not constitute a proof of the Prime Number Theorem, because the PNT is used
in the proof. Chebyshev (1850) also used his main results to prove Bertrand’s
postulate. Simpler proofs have been devised by various authors. For an easy
exposition, see Theorem 8.7 of Niven, Zuckerman & Montgomery (1991).
Richert (1949a, b) (cf. M¸ akowski 1960) used Bertrand’s postulate to show that
every integer> 6 can be expressed as a sum of distinct primes. Rosser &
Schoenfeld (1962, 1975) and Schoenfeld (1976) have given a large number of
very useful explicit estimates for primes and for the Chebyshev functions, of
which one example is thatπ(x ) > x /log x for all x ≥ 17. For the kth prime
number, pk , Dusart (1999) has given the lower bound
pk > k(log k + log log k − 1)
for k ≥ 2. For further explicit estimates, see Schoenfeld (1969), Costa Pereira
(1989), and Massias & Robin (1996). In Exercise 2.2.1 we find that ψ(x ) ≥
cx + O (1) with c = 1
2 log 5 = 0.8047 ... . This approach is mentioned by Gel’-
fond, in his editorial remarks in the Collected W orks of Chebyshev (1946, | {
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70 The elementary theory of arithmetic functions
pp. 285–288). Polynomials can be found that produce better constants, but
Gorshkov (1956) showed that the supremum of such constants is< 1, so
the Prime Number Theorem cannot be established by this method. For more
on this subject, see Montgomery (1994, Chapter 10), Pritsker (1999), and
Borwein (2002, Chapter 10).
Theorem 2.7(b)–(e) is due to Mertens (1874a, b). Our determination of the
constant in Theorem 2.7(e) incorporates an expository finesse due to Heath-
Brown.
Section 2.3. Theorem 2.9 is due to Landau (1903). Runge (1885) proved
(2.20), and Wigert (1906/7) showed that d (n) < n(log 2 +ε)/log log n for n > n0 (ε).
Ramanujan (1915a, b) established the upper bound of Theorem 2.11, first with
an extra log log logn in the error term, and then without. Ramanujan (1915b)
also proved that
log d (n)
log 2 < li(n) + O
(
n exp
(
− c
√
log n
))
for all n ≥ 2, and that
log d (n)
log 2 > li(n) + O
(
n exp
(
− c
√
log n
))
for infinitely many n. For a survey of extreme value estimates of arithmetic
functions, see Nicolas (1988).
Theorem 2.12 is due to Tur´ an (1934), although Corollary 2.13 and the es-
timate (2.22) used in the proof of Theorem 2.12 were established earlier by
Hardy & Ramanujan (1917). Kubilius (1956) generalized Tur´ an’s inequality to
arbitrary additive functions. See T enenbaum (1995, pp. 302–304) for a proof,
and discussion of the sharpest constants.
Theorem 2.14 is due to Hall & T enenbaum (1988, pp. 2, 11). It represents
a weakening of sharper estimates that can be derived with more work. For
example, Wirsing (1961) showed that iff is a multiplicative function such that
f (n) ≥ 0 for all n, if there is a constant C < 2 such that f ( pk ) ≪ C k for all
k ≥ 2, and if
∑
p≤x
f ( p) ∼ κx /log x
as x →∞ where κ is a positive real number, then
∑
n≤x
f (n) ∼ e−C0 κx
Ŵ(κ) log x
∏
p≤x
(
1 + f ( p)
p + f ( p2 )
p2 +···
)
.
For more information concerning non-negative multiplicative functions, see
Wirsing (1967), Hall (1974), Halberstam & Richert (1979), and Hildebrand | {
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