The Riemann Hypothesis (eBook)

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