Introduction to Analytic Number Theory

INTRODUCTION TO ANALYTIC NUMBER THEORY......Page 1
Title Page......Page 4
Copyright Page......Page 5
Preface......Page 6
Contents......Page 8
Historical Introduction......Page 14
1.1 Introduction......Page 26
1.3 Greatest common divisor......Page 27
1.4 Prime numbers......Page 29
1.5 The fundamental theorem of arithmetic......Page 30
1.6 The series of reciprocals of the primes......Page 31
1.7 The Euclidean algorithm......Page 32
1.8 The greatest common divisor of more than two numbers......Page 33
Exercises for Chapter 1......Page 34
2.2 The Möbius function μ(n)......Page 37
2.3 The Euler totient function φ(n)......Page 38
2.4 A relation connecting φ and μ......Page 39
2.5 A product formula for φ(n)......Page 40
2.6 The Dirichlet product of arithmetical functions......Page 42
2.7 Dirichlet inverses and the Möbius inversion formula......Page 43
2.8 The Mangoldt function Λ(n)......Page 45
2.9 Multiplicative functions......Page 46
2.10 Multiplicative functions and Dirichlet multiplication......Page 48
2.11 The inverse of a completely multiplicative function......Page 49
2.12 Liouville's function λ(n)......Page 50
2.13 The divisor functions σ α (n)......Page 51
2.14 Generalized convolutions......Page 52
2.15 Formal power series......Page 54
2.16 The Bell series of an arithmetical function......Page 55
2.17 Bell series and Dirichlet multiplication......Page 57
2.18 Derivatives of arithmetical functions......Page 58
Exercises for Chapter 2......Page 59
3.1 Introduction......Page 65
3.2 The big oh notation. Asymptotic equality of functions......Page 66
3.3 Euler's summation formula......Page 67
3.4 Some elementary asymptotic formulas......Page 68
3.5 The average order of d(n)......Page 70
3.6 The average order of the divisor functions σ α (n)......Page 73
3.7 The average order of φ(n)......Page 74
3.8 An application to the distribution of lattice points visible from the origin......Page 75
3.9 The average order of μ(n) and of Λ(n)......Page 77
3.10 The partial sums of a Dirichlet product......Page 78
3.11 Applications to μ(n) and Λ(n)......Page 79
3.12 Another identity for the partial sums of a Dirichlet product......Page 82
Exercises for Chapter 3......Page 83
4.1 Introduction......Page 87
4.2 Chebyshev's functions ψ(x) and θ(x)......Page 88
4.3 Relations connecting θ(x) and π(x)......Page 89
4.4 Some equivalent forms of the prime number theorem......Page 92
4.5 Inequalities for π(n) and p n......Page 95
4.6 Shapiro's Tauberian theorem......Page 98
4.7 Applications of Shapiro's theorem......Page 101
4.8 An asymptotic formula for the partial sums ∑ p≤x (1/p)......Page 102
4.9 The partial sums of the Möbius function......Page 104
4.10 Brief sketch of an elementary proof of the prime number theorem......Page 111
4.11 Selberg's asymptotic formula......Page 112
Exercises for Chapter 4......Page 114
5.1 Definition and basic properties of congruences......Page 119
5.2 Residue classes and complete residue systems......Page 122
5.3 Linear congruences......Page 123
5.4 Reduced residue systems and the Euler–Fermat theorem......Page 126
5.5 Polynomial congruences modulo p. Lagrange's theorem......Page 127
5.6 Applications of Lagrange's theorem......Page 128
5.7 Simultaneous linear congruences. The Chinese remainder theorem......Page 130
5.8 Applications of the Chinese remainder theorem......Page 131
5.9 Polynomial congruences with prime power moduli......Page 133
5.10 The principle of cross-classification......Page 136
5.11 A decomposition property of reduced residue systems......Page 138
Exercises for Chapter 5......Page 139
6.1 Definitions......Page 142
6.3 Elementary properties of groups......Page 143
6.4 Construction of subgroups......Page 144
6.5 Characters of finite abelian groups......Page 146
6.6 The character group......Page 148
6.7 The orthogonality relations for characters......Page 149
6.8 Dirichlet characters......Page 150
6.9 Sums involving Dirichlet characters......Page 153
6.10 The nonvanishing of L(1, χ) for real nonprincipal χ......Page 154
Exercises for Chapter 6......Page 156
7.1 Introduction......Page 159
7.2 Dirichtet's theorem for primes of the form 4n – 1 and 4n + 1......Page 160
7.3 The plan of the proof of Dirichlet's theorem......Page 161
7.4 Proof of Lemma 7.4......Page 163
7.5 Proof of Lemma 7.5......Page 164
7.6 Proof of Lemma 7.6......Page 165
7.8 Proof of Lemma 7.7......Page 166
7.9 Distribution of primes in arithmetic progressions......Page 167
Exercises for Chapter 7......Page 168
8.1 Functions periodic modulo k......Page 170
8.2 Existence of finite Fourier series for periodic arithmetical functions......Page 171
8.3 Ramanujan's sum and generalizations......Page 173
8.4 Multiplicative properties of the sums s k (n)......Page 175
8.5 Gauss sums associated with Dirichlet characters......Page 178
8.6 Dirichlet characters with nonvanishing Gauss sums......Page 179
8.7 Induced moduli and primitive characters......Page 180
8.8 Further properties of induced moduli......Page 181
8.10 Primitive characters and separable Gauss sums......Page 184
8.11 The finite Fourier series of the Dirichlet characters......Page 185
8.12 Pólya's inequality for the partial sums of primitive characters......Page 186
Exercises for Chapter 8......Page 188
9.1 Quadratic residues......Page 191
9.2 Legendre's symbol and its properties......Page 192
9.3 Evaluation of (–1|p) and (2|p)......Page 194
9.4 Gauss' lemma......Page 195
9.5 The quadratic reciprocity law......Page 198
9.6 Applications of the reciprocity law......Page 199
9.7 The Jacobi symbol......Page 200
9.8 Applications to Diophantine equations......Page 203
9.9 Gauss sums and the quadratic reciprocity law......Page 205
9.10 The reciprocity law for quadratic Gauss sums......Page 208
9.11 Another proof of the quadratic reciprocity law......Page 213
Exercises for Chapter 9......Page 214
10.1 The exponent of a number mod m. Primitive roots......Page 217
10.2 Primitive roots and reduced residue systems......Page 218
10.4 The existence of primitive roots mod p for odd primes p......Page 219
10.6 The existence of primitive roots mod p^α......Page 221
10.7 The existence of primitive roots mod 2p^α......Page 223
10.8 The nonexistence of primitive roots in the remaining cases......Page 224
10.9 The number of primitive roots mod m......Page 225
10.10 The index calculus......Page 226
10.11 Primitive roots and Dirichlet characters......Page 231
10.12 Real-valued Dirichlet characters mod p^α......Page 233
10.13 Primitive Dirichlet characters mod p^α......Page 234
Exercises for Chapter 10......Page 235
11.1 Introduction......Page 237
11.2 The half-plane of absolute convergence of a Dirichlet series......Page 238
11.3 The function defined by a Dirichlet series......Page 239
11.4 Multiplication of Dirichlet series......Page 241
11.5 Euler products......Page 243
11.6 The half-plane of convergence of a Dirichlet series......Page 245
11.7 Analytic properties of Dirichlet series......Page 247
11.8 Dirichlet series with nonnegative coefficients......Page 249
11.9 Dirichlet series expressed as exponentials of Dirichlet series......Page 251
11.10 Mean value formulas for Dirichlet series......Page 253
11.11 An integral formula for the coefficients of a Dirichlet series......Page 255
11.12 An integral formula for the partial sums of a Dirichlet series......Page 256
Exercises for Chapter 11......Page 259
12.1 Introduction......Page 262
12.2 Properties of the gamma function......Page 263
12.3 Integral representation for the Hurwitz zeta function......Page 264
12.4 A contour integral representation for the Hurwitz zeta function......Page 266
12.5 The analytic continuation of the Hurwitz zeta function......Page 267
12.6 Analytic continuation of ζ(s) and L(s, χ)......Page 268
12.7 Hurwitz's formula for ζ(s, a)......Page 269
12.8 The functional equation for the Riemann zeta function......Page 272
12.10 The functional equation for L-functions......Page 274
12.11 Evaluation of ζ(–n, a)......Page 277
12.12 Properties of Bernoulli numbers and Bernoulli polynomials......Page 278
12.14 Approximation of ζ(s, a) by finite sums......Page 281
12.15 Inequalities for |ζ(s, a)|......Page 283
12.16 Inequalities for |ζ(s)| and |L(s, χ)|......Page 285
Exercises for Chapter 12......Page 286
13.1 The plan of the proof......Page 291
13.2 Lemmas......Page 292
13.3 A contour integral representation for ψ 1 (x)/x²......Page 296
13.4 Upper bounds for |ζ(s)| and |ζ'(s)| near the line σ = 1......Page 297
13.5 The nonvanishing of ζ(s) on the line σ = 1......Page 299
13.6 Inequalities for |1/ζ(s)| and |ζ'(s)/|ζ(s)|......Page 300
13.7 Completion of the proof of the prime number theorem......Page 302
13.8 Zero-free regions for ζ(s)......Page 304
13.9 The Riemann hypothesis......Page 306
13.10 Application to the divisor function......Page 307
13.11 Application to Euler's totient......Page 310
13.12 Extension of Pólya's inequality for character sums......Page 312
Exercises for Chapter 13......Page 313
14.1 Introduction......Page 317
14.2 Geometric representation of partitions......Page 320
14.3 Generating functions for partitions......Page 321
14.4 Euler's pentagonal-number theorem......Page 324
14.5 Combinatorial proof of Euler's pentagonal-number theorem......Page 326
14.6 Euler's recursion formula for p(n)......Page 328
14.7 An upper bound for p(n)......Page 329
14.8 Jacobi's triple product identity......Page 331
14.9 Consequences of Jacobi's identity......Page 334
14.10 Logarithmic differentiation of generating functions......Page 335
14.11 The partition identities of Ramanujan......Page 337
Exercises for Chapter 14......Page 338
Bibliography......Page 342
Index of Special Symbols......Page 346
Index......Page 348
Back Cover......Page 352