This expanded and corrected second edition has a new chapter on the important topic of equidistribution. Undoubtedly, one cannot give an exhaustive treatment of the subject in a short chapter. How-ever, we hope that the problems presented here are enticing that the student will pursue further and learn from other sources.
A problem style presentation of the fundamental topics of ana-lytic number theory has its virtues, as I have heard from those who benefited from the first edition. Mere theoretical knowledge in any field is insufficient for a full appreciation of the subject and one of-ten needs to grapple with concrete questions in which these ideas are used in a vital way. Knowledge and the various layers of "know-ing" are difficult to define or describe. However, one learns much and gains insight only through practice. Making mistakes is an in-tegral part of learning. Indeed, "it is practice first and knowledge afterwards."

Preface to the Second Edition vii

Acknowledgments for the Second Edition ix

Preface to the First Edition xi

Acknowledgments for the First Edition xv

I Problems 1

1 Arithmetic Functions 3

      1.1 The Mobius Inversion Formula and Applications 4

      1.2 Formal Dirichlet Series 7

      1.3 Orders of Some Arithmetical Functions 9

      1.4 Average Orders of Arithmetical Functions 10

      1.5 Supplementary Problems 11

2 Primes in Arithmetic Progressions 17

      2.1 Summation Techniques 17

      2.2 Characters mod q 22

      2.3 Dirichlet's Theorem 24

      2.4 Contents Dirichlet's Hyperbola Method 27

      2.5 Supplementary Problems 29

3 The Prime Number Theorem 35

      3.1 Chebyshev's Theorem 36

      3.2 Nonvanishing of Dirichlet Series on Re(s) = 1 39

      3.3 The Ikehara - Wiener Theorem 42

      3.4 Supplementary Problems 48

4 The Method of Contour Integration 53

      4.1 Some Basic Integrals 53

      4.2 The Prime Number Theorem 57

      4.3 Further Examples 62

      4.4 Supplementary Problems 65

5 Functional Equations 69

      5.1 Poisson's Summation Formula 69

      5.2 The Riemann Zeta Function 72

      5.3 Gauss Sums 75

      5.4 Dirichlet L-functions 76

      5.5 Supplementary Problems 79

6 Hadamard Products 85

      6.1 Jensen's Theorem 85

      6.2 Entire Functions of Order 1 88

      6.3 The Gamma Function 91

      6.4 Infinite Products for ξ(s) and ξ(s, x) 93

      6.5 Zero-Free Regions for ζ(s) and L(s, x) 94

      6.6 Supplementary Problems 99

7 Explicit Formulas 101

      7.1 Counting Zeros 101

      7.2 Explicit Formula for Ψ(x) 104

      7.3 Weil's Explicit Formula 107

      7.4 Supplementary Problems 110

8 The Selberg Class 115

      8.1 The Phragmen - Lindelof Theorem 116

      8.2 Basic Properties 118

      8.3 Selberg's Conjectures 123

      8.4 Supplementary Problems 125

9 Sieve Methods 127

      9.1 The Sieve of Eratosthenes 127

      9.2 Brun's Elementary Sieve 133

      9.3 Selberg's Sieve 138

      9.4 Supplementary Problems 144

10 p-adic Methods 147

      10.1 Ostrowski's Theorem 147

      10.2 Hensel's Lemma 155

      10.3 p-adic Interpolation 159

      10.4 The p-adic Zeta-Function 165

      10.5 Supplementary Problems 168

11 Equidistribution 171

      11.1 Uniform distribution modulo 1 171

      11.2 Normal numbers 177

      11.3 Asymptotic distribution functions mod 1 180

      11.4 Discrepancy 182

      11.5 Equidistribution and L-functions 189

      11.6 Supplementary Problems 193

II Solutions 197

1 Arithmetic Functions 199

      1.1 The Mobius Inversion Formula and Applications 199

      1.2 Formal Dirichlet Series 208

      1.3 Orders of Some Arithmetical Functions 212

      1.4 Average Orders of Arithmetical Functions 215

      1.5 Supplementary Problems 220

2 Primes in Arithmetic Progressions 237

      2.1 Characters mod q 237

      2.2 Dirichlet's Theorem 246

      2.3 Dirichlet's Hyperbola Method 251

      2.4 Supplementary Problems 257

3 The Prime Number Theorem 273

      3.1 Chebyshev's Theorem 273

      3.2 Nonvanishing of Dirichlet Series on Re(s) = 1 280

      3.3 The Ikehara - Wiener Theorem 288

      3.4 Supplementary Problems 292

4 The Method of Contour Integration 305

      4.1 Some Basic Integrals 305

      4.2 The Prime Number Theorem 311

      4.3 Further Examples 314

      4.4 Supplementary Problems 317

5 Functional Equations 329

      5.1 Poisson's Summation Formula 329

      5.2 The Riemann Zeta Function 332

      5.3 Gauss Sums 333

      5.4 Dirichlet L-functions 335

      5.5 Supplementary Problems 338

6 Hadamard Products 357

      6.1 Jensen's theorem 357

      6.2 The Gamma Function 358

      6.3 Infinite Products for ξ(s) and ξ(s, x) 369

      6.4 Zero-Free Regions for ζ(s) and L(s, x) 374

      6.5 Supplementary Problems 379

7 Explicit Formulas 385

      7.1 Counting Zeros 385

      7.2 Explicit Formula for Ψ(x) 388

      7.3 Supplementary Problems 393

8 The Selberg Class 403

      8.1 The Phragmen - Lindelof Theorem 403

      8.2 Basic Properties 404

      8.3 Selberg's Conjectures 411

      8.4 Supplementary Problems 417

9 Sieve Methods 423

      9.1 The Sieve of Eratosthenes 423

      9.2 Brun's Elementary Sieve 429

      9.3 Selberg's Sieve 432

      9.4 Supplementary Problems 439

10 p-adic Methods 449

      10.1 Ostrowski's Theorem 449

      10.2 Hensel's Lemma 454

      10.3 p-adic Interpolation 456

      10.4 The p-adic ζ-Function 463

      10.5 Supplementary Problems 468

11 Equidistribution 475

      11.1 Uniform distribution medulo1 475

      11.2 Normal numbers 483

      11.3 Asymptotic distribution functions mod1 484

      11.4 Discrepancy 485

      11.5 Equidistribution and L-functions 488

      11.6 Supplementary Problem 490

References 497

Index 499

中科院文献情报中心