Introduction to Analytic and Probabilistic Number Theory

INTRODUCTION TO ANALYTIC AND PROBABILISTIC NUMBER THEORY......Page 1
Half-title......Page 2
Cambridge Studies in Advnaced Mathematics......Page 3
Title Page......Page 4
Copyright Page......Page 5
Dedication......Page 6
Contents......Page 8
Preface......Page 14
Notation......Page 16
Part I: Elementary methods......Page 18
§ 0.1 Abel summation......Page 20
§ 0.2 The Euler–Maclaurin summation formula......Page 22
Exercises......Page 24
§ 1.1 Introduction......Page 26
§ 1.2 Chebyshev's estimates......Page 27
§ 1.3 p-adic valuation of n!......Page 30
§ 1.4 Mertens' first theorem......Page 31
§ 1.5 Two new asymptotic formulae......Page 32
§ 1.6 Mertens' formula......Page 34
§ 1.7 Another theorem of Chebyshev......Page 36
Exercises......Page 37
§ 2.2 Examples......Page 40
§ 2.3 Formal Dirichlet series......Page 42
§ 2.4 The ring of arithmetic functions......Page 43
§ 2.5 The Möbius inversion formulae......Page 45
§ 2.6 Von Mangoldt's function......Page 47
§ 2.7 Euler's totient function......Page 49
Notes......Page 50
Exercises......Page 51
§ 3.2 Dirichlet's problem and the hyperbola method......Page 53
§ 3.4 Euler's totient function......Page 56
§ 3.5 The functions ω and Ω......Page 58
§ 3.6 Mean value of the Möbius function and the summatory functions of Chebyshev......Page 59
§ 3.7 Squarefree integers......Page 63
§ 3.8 Mean value of a multiplicative function with values in [0,1]......Page 65
Notes......Page 67
Exercises......Page 70
§ 4.1 The sieve of Eratosthenes......Page 73
§ 4.2 Brun's combinatorial sieve......Page 74
§ 4.3 Application to prime twins......Page 77
§ 4.4 The large sieve—analytic form......Page 79
§ 4.5 The large sieve—arithmetic form......Page 85
§ 4.6 Applications......Page 88
Notes......Page 91
Exercises......Page 93
§ 5.1 Introduction and definitions......Page 97
§ 5.2 The function τ(n)......Page 98
§ 5.3 The functions ω(n) and Ω(n)......Page 100
§ 5.4 Euler's function φ(n)......Page 101
§ 5.5 The functions σ κ (n), κ > 0......Page 102
Exercises......Page 104
§ 6.1 Introduction......Page 107
§ 6.2 Trigonometric integrals......Page 108
§ 6.3 Trigonometric sums......Page 109
§ 6.4 Application to the theorem of Voronoï......Page 113
Notes......Page 116
Exercises......Page 117
Part II: Methods of complex analysis......Page 120
§ 1.1 Convergent Dirichlet series......Page 122
§ 1.2 Dirichlet series of multiplicative functions......Page 123
§ 1.3 Fundamental analytic properties of Dirichlet series......Page 124
§ 1.4 Abscissa of convergence and mean value......Page 131
§ 1.5 An arithmetic application: the kernel of an integer......Page 133
§ 1.6 Order of magnitude in vertical strips......Page 135
Notes......Page 139
Exercises......Page 144
§ 2.1 Perron formulae......Page 147
§ 2.2 Application: a convergence theorem......Page 151
§ 2.3 The mean value formula......Page 153
Notes......Page 154
Exercises......Page 155
§ 3.2 Analytic continuation......Page 156
§ 3.3 Functional equation......Page 159
§ 3.4 Approximations and bounds in the critical strip......Page 160
§ 3.5 Initial localisation of zeros......Page 164
§ 3.6 Lemmas from complex analysis......Page 166
§ 3.7 Global distribution of zeros......Page 168
§ 3.8 Expansion as a Hadamard product......Page 172
§ 3.9 Zero-free regions......Page 174
§ 3.10 Bounds for ζ'/ζ, 1/ζ and log ζ......Page 175
Notes......Page 177
Exercises......Page 179
§ 4.1 The prime number theorem......Page 184
§ 4.2 Minimal hypotheses......Page 185
§ 4.3 The Riemann hypothesis......Page 187
Notes......Page 191
Exercises......Page 194
§ 5.1 Complex powers of ζ(s)......Page 197
§ 5.2 Hankel's formula......Page 200
§ 5.3 The main result......Page 201
§ 5.4 Proof of Theorem 3......Page 204
§ 5.5 A variant of the main theorem......Page 208
Notes......Page 212
Exercises......Page 214
§ 6.1 Integers having k prime factors......Page 217
§ 6.2 The average distribution of divisors: the arcsine law......Page 224
Notes......Page 229
Exercises......Page 231
§ 7.1 Introduction: Abelian/Tauberian theorems duality......Page 234
§ 7.2 Tauber's theorem......Page 237
§ 7.3 The theorems of Hardy–Littlewood and Karamata......Page 239
§ 7.4 The remainder term in Karamata's theorem......Page 244
§ 7.5 Ikehara's theorem......Page 251
§ 7.6 The Berry–Esseen inequality......Page 257
Notes......Page 259
Exercises......Page 261
§ 8.1 Introduction: Dirichlet characters......Page 265
§ 8.2 L-series. The prime number theorem for arithmetic progressions......Page 269
§ 8.3 Lower bounds for |L(s, χ)| when σ ≥ 1. Proof of Theorem 4......Page 273
Notes......Page 279
Exercises......Page 281
Part III: Probabilistic methods......Page 284
§ 1.1 Definitions. Natural density......Page 286
§ 1.2 Logarithmic density......Page 289
§ 1.3 Analytic density......Page 290
Notes......Page 292
Exercises......Page 293
§ 2.1 Definition—distribution functions......Page 298
§ 2.2 Characteristic functions......Page 302
Notes......Page 305
Exercises......Page 312
§ 3.1 Definition......Page 316
§ 3.2 The Turán–Kubilius inequality......Page 317
§ 3.3 Dual form of the Turán–Kubilius inequality......Page 321
§ 3.4 The Hardy–Ramanujan theorem and other applications......Page 322
§ 3.5 Effective mean value estimates for multiplicative functions......Page 325
§ 3.6 Normal structure of the set of prime factors of an integer......Page 328
Notes......Page 330
Exercises......Page 336
§ 4.1 The Erdős–Wintner theorem......Page 342
§ 4.2 Delange's theorem......Page 348
§ 4.3 Halász' theorem......Page 352
§ 4.4 The Erdős–Kac theorem......Page 364
Notes......Page 367
Exercises......Page 370
§ 5.1 Introduction. Rankin's method......Page 375
§ 5.2 The geometric method......Page 380
§ 5.3 Functional equations......Page 382
§ 5.4 Dickman's function......Page 387
§ 5.5 Approximations to Ψ(x, y) by the saddle-point method......Page 394
Notes......Page 404
Exercises......Page 408
§ 6.1 Introduction......Page 412
§ 6.2 Functional equations......Page 415
§ 6.3 Buchstab's function......Page 420
§ 6.4 Approximations to Φ(x, y) by the saddle-point method......Page 425
Notes......Page 435
Exercises......Page 437
Bibliography......Page 441
Index......Page 460
Back Cover......Page 466