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

Preface

Contents

Notation

Part I Introduction to the Riemann Hypothesis

1 Why This Book

1.1 The Holy Grail

1.2 Riemann’s Zeta and Liouville’s Lambda

1.3 The Prime Number Theorem

2 Analytic Preliminaries

2.1 The Riemann Zeta Function

2.2 Zero-free Region

2.3 Counting the Zeros of (s)

2.4 Hardy’s Theorem

3 Algorithms for Calculating (s)

3.1 Euler–MacLaurin Summation

3.2 Backlund

3.3 Hardy’s Function

3.4 The Riemann–Siegel Formula

3.5 Gram’s Law

3.6 Turing

3.7 The Odlyzko–Sch¨ onhage Algorithm

3.8 A Simple Algorithm for the Zeta Function

3.9 Further Reading

4 Empirical Evidence

4.1 Verification in an Interval

4.2 A Brief History of Computational Evidence

4.3 The Riemann Hypothesis and Random Matrices

4.4 The Skewes Number

5 Equivalent Statements

5.1 Number-Theoretic Equivalences

5.2 Analytic Equivalences

5.3 Other Equivalences

6 Extensions of the Riemann Hypothesis

6.1 The Riemann Hypothesis

6.2 The Generalized Riemann Hypothesis

6.3 The Extended Riemann Hypothesis

6.4 An Equivalent Extended Riemann Hypothesis

6.5 Another Extended Riemann Hypothesis

6.6 The Grand Riemann Hypothesis

7 Assuming the Riemann Hypothesis and Its Extensions . . .

7.1 Another Proof of The Prime Number Theorem

7.2 Goldbach’s Conjecture

7.3 More Goldbach

7.4 Primes in a Given Interval

7.5 The Least Prime in Arithmetic Progressions

7.6 Primality Testing

7.7 Artin’s Primitive Root Conjecture

7.8 Bounds on Dirichlet L-Series

7.9 The Lindel¨ of Hypothesis

7.10 Titchmarsh’s S( T ) Function

7.11 Mean Values of (s)

8 Failed Attempts at Proof

8.1 Stieltjes and Mertens’ Conjecture

8.2 Hans Rademacher and False Hopes

8.3 Tur´ an’s Condition

8.4 Louis de Branges’s Approach

8.5 No Really Good Idea

9 Formulas

10 Timeline

Part II Original Papers

11 Expert Witnesses

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

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

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

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

12 The Experts Speak for Themselves

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

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

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

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

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

12.6 G. H. Hardy (1915) Prime Numbers

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

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

12.9 P. Turan (1948)

12.10 A. Selberg (1949)

12.11 P. Erdös (1949)

12.12 S. Skewes (1955)

12.13 C. B. Haselgrove (1958)

12.14 H. Montgomery (1973)

12.15 D. J. Newman (1980)

12.16 J. Korevaar (1982)

12.17 H. Daboussi (1984)

12.18 A. Hildebrand (1986)

12.19 D. Goldston and H. Montgomery (1987)

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

References

Index

Index