Riemann's Zeta Function

Superb study of one of the most influential classics in mathematics examines the landmark 1859 publication entitled "On the Number of Primes Less Than a Given Magnitude," and traces developments in theory inspired by it. Topics include Riemann's main formula, the prime number theorem, the Riemann-Siegel formula, large-scale computations, Fourier analysis, and other related topics.

Preface

ix

Acknowledgments

xiii

Riemann's Paper

The Historical Context of the Paper

1

(5)

The Euler Product Formula

6

(1)

The Factorial Function

7

(2)

The Function ζ(s)

9

(2)

Values of ζ(s)

11

(1)

First Proof of the Functional Equation

12

(3)

Second Proof of the Functional Equation

15

(1)

The Function ζ(s)

16

(2)

The Roots p of ζ

18

(2)

The Product Representation of ζ(s)

20

(2)

The Connection between ζ(s) and Primes

22

(1)

Fourier Inversion

23

(2)

Method for Deriving the Formula for J(x)

25

(1)

The Principal Term of J(x)

26

(3)

The Term Involving the Roots p

29

(2)

The Remaining Terms

31

(2)

The Formula for π(x)

33

(3)

The Density dJ

36

(1)

Questions Unresolved by Riemann

37

(2)

The Product Formula for ζ

Introduction

39

(1)

Jensen's Theorem

40

(1)

A Simple Estimate of ζ(s)

41

(1)

The Resulting Estimate of the Roots p

42

(1)

Convergence of the Product

42

(1)

Rate of Growth of the Quotient

43

(2)

Rate of Growth of Even Entire Functions

45

(1)

The Product Formula for ζ

46

(2)

Riemann's Main Formula

Introduction

48

(2)

Derivation of von Mangoldt's Formula for ψ(x)

50

(4)

The Basic Integral Formula

54

(2)

The Density of the Roots

56

(2)

Proof of von Mangoldt's Formula for ψ(x)

58

(3)

Riemann's Main Formula

61

(1)

Von Mangoldt's Proof of Riemann's Main Formula

62

(4)

Numerical Evaluation of the Constant

66

(2)

The Prime Number Theorem

Introduction

68

(2)

Hadamard's Proof that re p< 1 for All p

70

(2)

Proof That ψ(x) ∼ x

72

(4)

Proof of the Prime Number Theorem

76

(2)

De la Vallee Poussin's Theorem

Introduction

78

(1)

An Improvement of Re p < 1

79

(2)

De la Vallee Poussin's Estimate of the Error

81

(3)

Other Formulas for π(x)

84

(4)

Error Estimates and the Riemann Hypothesis

88

(3)

A Postscript to de la Vallee Poussin's Proof

91

(5)

Numerical Analysis of the Roots by Euler-Maclaurin Summation

Introduction

96

(2)

Euler-Maclaurin Summation

98

(8)

Evaluation of II by Euler-Maclaurin Summation. Stirling's Series

106

(8)

Evaluation of ζ by Euler-Maclaurin Summation

114

(5)

Techniques for Locating Roots on the Line

119

(8)

Techniques for Computing the Number of Roots in a Given Range

127

(5)

Backlund's Estimate of N (T)

132

(2)

Alternative Evaluation of ζ(0)/ζ(0)

134

(2)

The Riemann-Siegel Formula

Introduction

136

(1)

Basic Derivation of the Formula

137

(4)

Estimation of the Integral away from the Saddle Point

141

(4)

First Approximation to the Main Integral

145

(3)

Higher Order Approximations

148

(7)

Sample Computations

155

(7)

Error Estimates

162

(2)

Speculations on the Genesis of the Riemann Hypothesis

164

(2)

The Riemann-Siegel Integral Formula

166

(5)

Large-Scale Computations

Introduction

171

(1)

Turing's Method

172

(3)

Lehmer's Phenomenon

175

(4)

Computations of Rosser, Yohe, and Schoenfeld

179

(3)

The Growth of Zeta as t → and the Location of Its Zeros

Introduction

182

(1)

Lindelof's Estimates and His Hypothesis

183

(4)

The Three Circles Theorem

187

(1)

Backlund's Reformulation of the Lindel of Hypothesis

188

(2)

The Average Value of S(t) Is Zero

190

(3)

The Bohr-Landau Theorem

193

(2)

The Average of ζ(s)

195

(4)

Further Results, Landau's Notation o, O

199

(4)

Fourier Analysis

Invariant Operators on R+ and Their Transforms

203

(2)

Adjoints and Their Transforms

205

(1)

A Self-Adjoint Operator with Transform ζ(s)

206

(3)

The Functional Equation

209

(3)

2ζ(s)/s(s--1) as a Transform

212

(1)

Fourier Inversion

213

(2)

Parseval's Equation

215

(1)

The Values of ζ(--n)

216

(1)

Mobius Inversion

217

(1)

Ramanujan's Formula

218

(8)

Zeros on the Line

Hardy's Theorem

226

(3)

There Are at Least KT Zeros on the Line

229

(8)

There Are at Least KT log T Zeros on the Line

237

(9)

Proof of a Lemma

246

(14)

Miscellany

The Riemann Hypothesis and the Growth of M(x)

260

(3)

The Riemann Hypothesis and Farey Series

263

(5)

Denjoy's Probabilistic Interpretation of the Riemann Hypothesis

268

(1)

An Interesting False Conjecture

269

(1)

Transforms with Zeros on the Line

269

(4)

Alternative Proof of the Integral Formula

273

(5)

Tauberian Theorems

278

(3)

Chebyshev's Identity

281

(3)

Selberg's Inequality

284

(4)

Elementary Proof of the Prime Number Theorem

288

(10)

Other Zeta Functions. Weil's Theorem

298

(1)

Appendix On the Number of Primes Less Than a Given Magnitude

299

(7)

Bernhard Riemann

References

306

(5)

Index

311

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