The Riemann hypothesis : a resource for the afficionado and virtuoso alike / P. Borwein ... [and others].

Format:

Book

Contributor:

Borwein, Peter B.

Anne and Joseph Trachtman Memorial Book Fund.

Series:

CMS books in mathematics

Language:

English

Subjects (All):

Riemann hypothesis.

Numbers, Prime.

Number theory.

Riemann, Bernhard, 1826-1866.

Riemann, Bernhard.

Physical Description:

xiv, 533 pages : illustrations ; 24 cm.

Place of Publication:

New York : Springer, 2008.

Summary:

The Riemann Hypothesis has become the Holy Grail of mathematics in the century and a half since 1859 when Bernhard Riemann, one of the extraordinary mathematical talents of the 19th century, originally posed the problem. While the problem is notoriously difficult, and complicated even to state carefully, it can be loosely formulated as "the number of integers with an even number of prime factors is the same as the number of integers with an odd number of prime factors." The Hypothesis makes a very precise connection between two seemingly unrelated mathematical objects, namely prime numbers and the zeros of analytic functions. If solved, it would give us profound insight into number theory and, in particular, the nature of prime numbers.

This book is an introduction to the theory surrounding the Riemann Hypothesis. Part I serves as a compendium of known results and as a primer for the material presented in the 20 original papers contained in Part II. The original papers place the material into historical context and illustrate the motivations for research on and around the Riemann Hypothesis. Several of these papers focus on computation of the zeta function, while others give proofs of the Prime Number Theorem, since the Prime Number Theorem is so closely connected to the Riemann Hypothesis. The text is suitable for a graduate course or seminar or simply as a reference for anyone interested in this extraordinary conjecture.

Contents:

Part I Introduction to the Riemann Hypothesis

1 Why This Book 3

1.1 The Holy Grail 3

1.2 Riemann's Zeta and Liouville's Lambda 5

1.3 The Prime Number Theorem 7

2 Analytic Preliminaries 9

2.1 The Riemann Zeta Function 9

2.2 Zero-free Region 16

2.3 Counting the Zeros of [zeta](s) 18

2.4 Hardy's Theorem 24

3 Algorithms for Calculating [zeta](s) 29

3.1 Euler-MacLaurin Summation 29

3.2 Backlund 30

3.3 Hardy's Function 31

3.4 The Riemann-Siegel Formula 32

3.5 Gram's Law 33

3.6 Turing 34

3.7 The Odlyzko-Schonhage Algorithm 35

3.8 A Simple Algorithm for the Zeta Function 35

4 Empirical Evidence 37

4.1 Verification in an Interval 37

4.2 A Brief History of Computational Evidence 39

4.3 The Riemann Hypothesis and Random Matrices 40

4.4 The Skewes Number 43

5 Equivalent Statements 45

5.1 Number-Theoretic Equivalences 45

5.2 Analytic Equivalences 49

5.3 Other Equivalences 52

6 Extensions of the Riemann Hypothesis 55

6.1 The Riemann Hypothesis 55

6.2 The Generalized Riemann Hypothesis 56

6.3 The Extended Riemann Hypothesis 57

6.4 An Equivalent Extended Riemann Hypothesis 57

6.5 Another Extended Riemann Hypothesis 58

6.6 The Grand Riemann Hypothesis 58

7 Assuming the Riemann Hypothesis and Its Extensions 61

7.1 Another Proof of The Prime Number Theorem 61

7.2 Goldbach's Conjecture 62

7.3 More Goldbach 62

7.4 Primes in a Given Interval 63

7.5 The Least Prime in Arithmetic Progressions 63

7.6 Primality Testing 63

7.7 Artin's Primitive Root Conjecture 64

7.8 Bounds on Dirichlet L-Series 64

7.9 The Lindelof Hypothesis 65

7.10 Titchmarsh's S(T) Function 65

7.11 Mean Values of [zeta](s) 66

8 Failed Attempts at Proof 69

8.1 Stieltjes and Mertens' Conjecture 69

8.2 Hans Rademacher and False Hopes 70

8.3 Turan's Condition 71

8.4 Louis de Branges's Approach 71

8.5 No Really Good Idea 72

9 Formulas 73

Part II Original Papers

11 Expert Witnesses 93

11.1 E. Bombieri (2000-2001) 94

11.2 P. Sarnak (2004) 106

11.3 J. B. Conrey (2003) 116

11.4 A. Ivic (2003) 130

12 The Experts Speak for Themselves 161

12.1 P. L. Chebyshev (1852) 162

12.2 B. Riemann (1859) 183

12.3 J. Hadamard (1896) 199

12.4 C. de la Vallee Poussin (1899) 222

12.5 G. H. Hardy (1914) 296

12.6 G. H. Hardy (1915) 300

12.7 G. H. Hardy and J. E. Littlewood (1915) 307

12.8 A. Weil (1941) 313

12.9 P. Turan (1948) 317

12.10 A. Selberg (1949) 353

12.11 P. Erdos (1949) 363

12.12 S. Skewes (1955) 375

12.13 C. B. Haselgrove (1958) 399

12.14 H. Montgomery (1973) 405

12.15 D. J. Newman (1980) 419

12.16 J. Korevaar (1982) 424

12.17 H. Daboussi (1984) 433

12.18 A. Hildebrand (1986) 438

12.19 D. Goldston and H. Montgomery (1987) 447

12.20 M. Agrawal, N. Kayal, and N. Saxena (2004) 469.

Notes:

Includes bibliographical references and index.

Local Notes:

Acquired for the Penn Libraries with assistance from the Anne and Joseph Trachtman Memorial Book Fund.

ISBN:

9780387721255

0387721258

OCLC:

190760103

Online:

Publisher description