Multiplicative Number Theory I: Classical

Prime numbers are the multiplicative building blocks of natural numbers. Understanding their overall influence and especially their distribution gives rise to central questions in mathematics and physics. In particular, their finer distribution is closely connected with the Riemann hypothesis, the most important unsolved problem in the mathematical world. This book comprehensively covers all the topics met in first courses on multiplicative number theory and the distribution of prime numbers. The text is based on courses taught successfully over many years at the University of Michigan, Imperial College, London and Pennsylvania State University.

Robert Vaughan is a Professor of Mathematics at Pennsylvania State University.

Preface

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 Ω(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 Twin 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

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 Tauberian 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

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 formulae

397

12.1 Classical formulae

397

12.2 Weil'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

C The gamma function

520

C.1 Notes

531

C.2 References

533

D Topics 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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Multiplicative Number Theory I: Classical Theory

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Author Biography

Table of Contents

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