The Riemann Hypothesis - A Resource for the Afficionado and Virtuoso Alike

von: Peter Borwein, Stephen Choi, Brendan Rooney, Andrea Weirathmueller

Springer-Verlag, 2007

ISBN: 9780387721262 , 533 Seiten

Format: PDF

Kopierschutz: Wasserzeichen

Windows PC,Mac OSX Apple iPad, Android Tablet PC's

Preis: 107,09 EUR

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Mehr zum Inhalt

The Riemann Hypothesis - A Resource for the Afficionado and Virtuoso Alike


 

Preface

7

Contents

9

Notation

12

Part I Introduction to the Riemann Hypothesis

14

1 Why This Book

15

1.1 The Holy Grail

15

1.2 Riemann’s Zeta and Liouville’s Lambda

17

1.3 The Prime Number Theorem

19

2 Analytic Preliminaries

21

2.1 The Riemann Zeta Function

21

2.2 Zero-free Region

28

2.3 Counting the Zeros of (s)

30

2.4 Hardy’s Theorem

36

3 Algorithms for Calculating (s)

40

3.1 Euler–MacLaurin Summation

40

3.2 Backlund

41

3.3 Hardy’s Function

42

3.4 The Riemann–Siegel Formula

43

3.5 Gram’s Law

44

3.6 Turing

45

3.7 The Odlyzko–Sch¨ onhage Algorithm

46

3.8 A Simple Algorithm for the Zeta Function

46

3.9 Further Reading

47

4 Empirical Evidence

48

4.1 Verification in an Interval

48

4.2 A Brief History of Computational Evidence

50

4.3 The Riemann Hypothesis and Random Matrices

51

4.4 The Skewes Number

54

5 Equivalent Statements

56

5.1 Number-Theoretic Equivalences

56

5.2 Analytic Equivalences

60

5.3 Other Equivalences

63

6 Extensions of the Riemann Hypothesis

66

6.1 The Riemann Hypothesis

66

6.2 The Generalized Riemann Hypothesis

67

6.3 The Extended Riemann Hypothesis

68

6.4 An Equivalent Extended Riemann Hypothesis

68

6.5 Another Extended Riemann Hypothesis

69

6.6 The Grand Riemann Hypothesis

69

7 Assuming the Riemann Hypothesis and Its Extensions . . .

72

7.1 Another Proof of The Prime Number Theorem

72

7.2 Goldbach’s Conjecture

73

7.3 More Goldbach

73

7.4 Primes in a Given Interval

74

7.5 The Least Prime in Arithmetic Progressions

74

7.6 Primality Testing

74

7.7 Artin’s Primitive Root Conjecture

75

7.8 Bounds on Dirichlet L-Series

75

7.9 The Lindel¨ of Hypothesis

76

7.10 Titchmarsh’s S( T ) Function

76

7.11 Mean Values of (s)

77

8 Failed Attempts at Proof

79

8.1 Stieltjes and Mertens’ Conjecture

79

8.2 Hans Rademacher and False Hopes

80

8.3 Tur´ an’s Condition

81

8.4 Louis de Branges’s Approach

81

8.5 No Really Good Idea

82

9 Formulas

83

10 Timeline

90

Part II Original Papers

100

11 Expert Witnesses

101

11.1 E. Bombieri (2000–2001) Problems of the Millennium: The Riemann Hypothesis

102

11.2 P. Sarnak (2004) Problems of the Millennium: The Riemann Hypothesis

114

11.3 J. B. Conrey (2003) The Riemann Hypothesis

124

11.4 A. Ivi ´ c (2003) On Some Reasons for Doubting the Riemann Hypothesis

138

12 The Experts Speak for Themselves

169

12.1 P. L. Chebyshev (1852) Sur la fonction qui d ´ etermine la totalit ´ e des nombres premiers inf ´ erieurs ` a une limite donn ´ ee

170

12.2 B. Riemann (1859) Ueber die Anzahl der Primzahlen unter einer gegebe-nen Gr ¨ osse

191

12.3 J. Hadamard (1896) Sur la distribution des z ´ eros de la fonction (s) et ses cons ´ equences arithm ´ etiques

207

12.4 C. de la Vall ´ ee Poussin (1899) Sur la fonction ( s) de Riemann et le nombre des nom-bres premiers inf ´ erieurs a une limite donn ´ ee

230

12.5 G. H. Hardy (1914) Sur les z ´ eros de la fonction (s) de Riemann

304

12.6 G. H. Hardy (1915) Prime Numbers

308

12.7 G. H. Hardy and J. E. Littlewood (1915) New Proofs of the Prime- Number Theorem and Simi-lar Theorems

315

12.8 A. Weil (1941) On the Riemann hypothesis in Function-Fields

321

12.9 P. Turan (1948)

325

12.10 A. Selberg (1949)

361

12.11 P. Erdös (1949)

371

12.12 S. Skewes (1955)

383

12.13 C. B. Haselgrove (1958)

407

12.14 H. Montgomery (1973)

413

12.15 D. J. Newman (1980)

427

12.16 J. Korevaar (1982)

432

12.17 H. Daboussi (1984)

441

12.18 A. Hildebrand (1986)

446

12.19 D. Goldston and H. Montgomery (1987)

455

12.20 M. Agrawal, N. Kayal, and N. Saxena (2004)

477

References

491

Index

509

Index

509