Tom M. Apostol
Introduction
to Analytic
Number Theory
I
Springer-Verlag
New York
1976
Heidelberg
Berlin
Contents
Historical Introduction
Chapter 1
The Fundamental Theorem of Arithmetic
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
Introduction 13
Divisibility 14
Greatest common divisor 14
Prime numbers 16
The fundamental theorem of arithmetic 17
The series of reciprocals of the primes 18
The Euclidean algorithm / 9
The greatest common divisor of more than two numbers 20
Exercises for Chapter 1 21
Chapter 2
Arithmetical Functions and Dirichlet Multiplication
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
2.9
2.10
2.11
Introduction 24
The Mobius function n(n) 24
The Euler totient function q>(n) 25
A relation connecting cp and ji 26
A product formula for <p(n) 27
The Dirichlet product of arithmetical functions 29
Dirichlet inverses and the Mobius inversion formula 30
The Mangoldt function A(n) -32
Multiplicative functions 33
Multiplicative functions and Dirichlet multiplication 35
The inverse of a completely multiplicative function 36
vn
2.12
2.13
2.14
2.15
2.16
2.17
2.18
2.19
Liouville's function X(n) 37
The divisor functions <T5(n) 38
Generalized convolutions 39
Formal power series 41
The Bell series of an arithmetical function 42
Bell series and Dirichlet multiplication 44
Derivatives of arithmetical functions 45
The Selberg identity 46
Exercises for Chapter 2 46
Chapter 3
Averages of Arithmetical Functions
3.1
3.2
3.3
3.4
3.5
3.6
3.7
3.8
3.9
3.10
3.11
3.12
Introduction 52
The big oh notation. Asymptotic equality of functions 53
Euler's summation formula 54
Some elementary asymptotic formulas 55
The average order of d(ri) 57
The average order of the divisor functions oj.n) 60
The average order of cp(n) 61
An application to the distribution of lattice points visible from the origin 62
The average order of fi(n) and of A(n) 64
The partial sums of a Dirichlet product 65
Applications to pin) and Mn) 66
Another identity for the partial sums of a Dirichlet product 69
Exercises for Chapter 3 70
Chapter 4
Some Elementary Theorems on the Distribution of Prime
Numbers
4=1
4.2
4.3
4.4
4.5
4.6
4.7
4.8
4.9
4.10
4.11
Introduction 74
Chebyshev's functions \j/(x) and B(x) • 75
Relations connecting 9(x) and n(x) 76
Some equivalent forms of the prime number theorem 79
Inequalities for n{n) and pn 82
Shapiro's Tauberian theorem 85
Applications of Shapiro's theorem 88
An asymptotic formula for the partial sums Y,p<x 0/p) #9
The partial sums of the Mobius function 91
Brief sketch of an elementary proof of the prime number theorem 98
Selberg's asymptotic formula 99
Exercises for Chapter 4 101
Chapter 5
Congruences
5.1
5.2
5.3
viii
Definition and basic properties of congruences 106
Residue classes and complete residue systems 109
Linear congruences 110
5.4
5.5
5.6
5.7
5.8
5.9
5.10
5.11
Reduced residue systems and the Euler-Fermat theorem 113
Polynomial congruences modulo p. Lagrange's theorem 114
Applications of Lagrange's theorem 115
Simultaneous linear congruences. The Chinese remainder theorem 117
Applications of the Chinese remainder theorem 118
Polynomial congruences with prime power moduli 120
The principle of cross-classification 123
A decomposition property of reduced residue systems 125
Exercises for Chapter 5 126
Chapter 6
Finite Abelian Groups and Their Characters
6.1
6.2
6.3
6.4
6.5
6.6
6.7
6.8
6.9
6.10
Definitions 129
Examples of groups and subgroups 130
Elementary properties of groups 130
Construction of subgroups 131
Characters of finite abelian groups 133
The character group 135
The orthogonality relations for characters 136
Dirichlet characters 137
Sums involving Dirichlet characters 140
The nonvanishing of L(l, /) for real nonprincipal % 141
Exercises for Chapter 6 143
Chapter 7
Dirichlet's Theorem on Primes in Arithmetic Progressions
7.1
7.2
7.3
l.A
7.5
7.6
1.1
7.8
7.9
Introduction 146
Dirichlet's theorem for primes of the form An — 1 and An + 1 147
The plan of the proof of Dirichlet's theorem 148
Proof of Lemma 7.4 150
Proof of Lemma 7.5 151
Proof of Lemma 7.6 152
Proof of Lemma 7.8 153 \
Proof of Lemma 7.7 153
Distribution of primes in arithmetic progressions 154
Exercises for Chapter 7 155
Chapter 8
Periodic Arithmetical Functions and Gauss Sums
8.1 Functions periodic modulo k 157
8.2 Existence of finite Fourier series for periodic arithmetical functions
8.3 Ramanujan's sum and generalizations 160
8.4 Multiplicative properties of the sums sk(n) 162
8.5 Gauss sums associated with Dirichlet characters 165
8.6 Dirichlet characters with nonvanishing Gauss sums 166
8.7 Induced moduli and primitive characters 167
158
IX
8.8
8.9
8.10
8.11
8.12
Further properties of induced moduli 768
The conductor of a character 171
Primitive characters and separable Gauss sums 171
The finite Fourier series of the Dirichlet characters 172
Polya's inequality for the partial sums of primitive characters
Exercises for Chapter 8 175
173
Chapter 9
Quadratic Residues and the Quadratic Reciprocity Law
9.1
9.2
9.3
9.4
9.5
9.6
9.7
9.8
9.9
9.10
9.11
Quadratic residues 178
Legendre's symbol and its properties 179
Evaluation of ( - 1 \p) and (2\p) 181
Gauss' lemma 182
The quadratic reciprocity law 185
Applications of the reciprocity law 186
The Jacobi symbol 187
Applications to Diophantine equations 190
Gauss sums and the quadratic reciprocity law 192
The reciprocity law for quadratic Gauss sums 195
Another proof of the quadratic reciprocity law 200
Exercises for Chapter 9 201
Chapter 10
Primitive Roots
10.1
10.2
10.3
10.4
10.5
10.6
10.7
10.8
10.9
10.10
10.11
10.12
10.13
The exponent of a number mod m. Primitive roots 204
Primitive roots and reduced residue systems 205
The nonexistence of primitive roots mod 2" for a > 3 206
The existence of primitive roots mod p for odd primes p 206
Primitive roots and quadratic residues 208
The existence of primitive roots mod p" 208
The existence of primitive roots mod 2p" 210
The nonexistence of primitive roots in the remaining cases 211
The number of primitive roots mod m 212
The index calculus 213
Primitive roots and Dirichlet characters 218
Real-valued Dirichlet characters mod p" 220
Primitive Dirichlet characters mod p" 221
Exercises for Chapter 10 222
Chapter 11
Dirichlet Series and Euler Products
11.1 Introduction 224
11.2 The half-plane of absolute convergence of a Dirichlet series 225
11.3 The function defined by a Dirichlet series 226
11.4
11.5
11.6
11.7
11.8
11.9
11.10
11.11
11.12
Multiplication of Dirichlet series 228
Euler products 230
The half-plane of convergence of a Dirichlet series 232
Analytic properties of Dirichlet series 234
Dirichlet series with nonnegative coefficients 236
Dirichlet series expressed as exponentials of Dirichlet series 238
Mean value formulas for Dirichlet series 240
An integral formula for the coefficients of a Dirichlet series 242
An integral formula for the partial sums of a Dirichlet series 243
Exercises for Chapter 11 246
Chapter 12
The Functions ((.?) and L(s,
12.1
12.2
12.3
12.4
12.5
12.6
12.7
12.8
12.9
12.10
12.11
12.12
12.13
12.14
12.15
12.16
•/)
Introduction 249
Properties of the gamma function 250
Integral representation for the Hurwitz zeta function 251
A contour integral representation for the Hurwitz zeta function 253
The analytic continuation of the Hurwitz zeta function 254
Analytic continuation of ((s) and L{s, x) 255
Hurwitz's formula for ((s, a) 256
The functional equation for the Riemann zeta function 259
A functional equation for the Hurwitz zeta function 261
The functional equation for L-functions 261
Evaluation of ((— n, a) 264
Properties of Bernoulli numbers and Bernoulli polynomials 265
Formulas for L(0, x) 268
Approximation of ((s, a) by finite sums 268
Inequalities for | t(s, a) \ 270
Inequalities for | C(s) | and | L(s, x) I 272
Exercises for Chapter 12 273
Chapter 13
Analytic Proof of the Prime Number Theorem
13.1
13.2
13.3
13.4
13.5
13.6
13.7
13.8
13.9
13.10
13.11
13.12
The plan of the proof 278
Lemmas 279
A contour integral representation for i/^x)/* 2 283
Upper bounds for |{(s)| and |('(s)| near the line a = 1 284
The nonvanishing of C(s) on the line u = 1 286
Inequalities for 11/C(s) | and | C(s)/t(s) | 287
Completion of the proof of the prime number theorem 289
Zero-free regions for C(s) 291
The Riemann hypothesis 293
Application to the divisor function 294
Application to Euler's totient 297
Extension of Polya's inequality for character sums 299
Exercises for Chapter 13 300
XI
Chapter 14
Partitions
14.1
14.2
14.3
14.4
14.5
14.6
I A.I
14.8
14.9
14.10
14.11
Introduction 304
Geometric representation of partitions 307
Generating functions for partitions 308
Euler's pentagonal-number theorem 311
Combinatorial proof of Euler's pentagonal-number theorem
Euler's recursion formula for p(n) 315
An upper bound for p(n) 316
Jacobi's triple product identity 378
Consequences of Jacobi's identity 327
Logarithmic differentiation of generating functions 322
The partition identities of Ramanujan 324
Exercises for Chapter 14 325
Bibliography 329
Index of Special Symbols 333
Index 335
XII
313