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Autor: Peter Borwein, Stephen Choi, Brendan Rooney, Andrea Weirathmueller
Herausgeber: Peter Borwein, Stephen Choi, Brendan Rooney, Andrea Weirathmueller
Titel: The Riemann Hypothesis A Resource for the Afficionado and Virtuoso Alike
Verlag: Springer-Verlag
ISBN/ISSN: 9780387721262
Auflage: 1
Preis : CHF 110.90
Erscheinungsdatum:
Inhalt
Kategorie: Wirtschaft/Management
Sprache: English
Technische Daten
Seiten: 533
Kopierschutz: DRM
Geräte: PC/MAC/eReader/Tablet
Formate: PDF
Inhaltsangabe

This book presents the Riemann Hypothesis, connected problems, and a taste of the body of theory developed towards its solution. It is targeted at the educated non-expert. Almost all the material is accessible to any senior mathematics student, and much is accessible to anyone with some university mathematics. The appendices include a selection of original papers that encompass the most important milestones in the evolution of theory connected to the Riemann Hypothesis. The appendices also include some authoritative expository papers. These are the 'expert witnesses' whose insight into this field is both invaluable and irreplaceable.

Inhaltsverzeichnis
Preface7
Contents9
Notation12
Part I Introduction to the Riemann Hypothesis14
1 Why This Book15
1.1 The Holy Grail15
1.2 Riemanns Zeta and Liouvilles Lambda17
1.3 The Prime Number Theorem19
2 Analytic Preliminaries21
2.1 The Riemann Zeta Function21
2.2 Zero-free Region28
2.3 Counting the Zeros of (s)30
2.4 Hardys Theorem36
3 Algorithms for Calculating (s)40
3.1 EulerMacLaurin Summation40
3.2 Backlund41
3.3 Hardys Function42
3.4 The RiemannSiegel Formula43
3.5 Grams Law44
3.6 Turing45
3.7 The OdlyzkoSch¨ onhage Algorithm46
3.8 A Simple Algorithm for the Zeta Function46
3.9 Further Reading47
4 Empirical Evidence48
4.1 Verification in an Interval48
4.2 A Brief History of Computational Evidence50
4.3 The Riemann Hypothesis and Random Matrices51
4.4 The Skewes Number54
5 Equivalent Statements56
5.1 Number-Theoretic Equivalences56
5.2 Analytic Equivalences60
5.3 Other Equivalences63
6 Extensions of the Riemann Hypothesis66
6.1 The Riemann Hypothesis66
6.2 The Generalized Riemann Hypothesis67
6.3 The Extended Riemann Hypothesis68
6.4 An Equivalent Extended Riemann Hypothesis68
6.5 Another Extended Riemann Hypothesis69
6.6 The Grand Riemann Hypothesis69
7 Assuming the Riemann Hypothesis and Its Extensions . . .72
7.1 Another Proof of The Prime Number Theorem72
7.2 Goldbachs Conjecture73
7.3 More Goldbach73
7.4 Primes in a Given Interval74
7.5 The Least Prime in Arithmetic Progressions74
7.6 Primality Testing74
7.7 Artins Primitive Root Conjecture75
7.8 Bounds on Dirichlet L-Series75
7.9 The Lindel¨ of Hypothesis76
7.10 Titchmarshs S( T ) Function76
7.11 Mean Values of (s)77
8 Failed Attempts at Proof79
8.1 Stieltjes and Mertens Conjecture79
8.2 Hans Rademacher and False Hopes80
8.3 Tur´ ans Condition81
8.4 Louis de Brangess Approach81
8.5 No Really Good Idea82
9 Formulas83
10 Timeline90
Part II Original Papers100
11 Expert Witnesses101
11.1 E. Bombieri (20002001) Problems of the Millennium: The Riemann Hypothesis102
11.2 P. Sarnak (2004) Problems of the Millennium: The Riemann Hypothesis114
11.3 J. B. Conrey (2003) The Riemann Hypothesis124
11.4 A. Ivi ´ c (2003) On Some Reasons for Doubting the Riemann Hypothesis138
12 The Experts Speak for Themselves169
12.1 P. L. Chebyshev (1852) Sur la fonction qui d ´ etermine la totalit ´ e des nombres premiers inf ´ erieurs ` a une limite donn ´ ee170
12.2 B. Riemann (1859) Ueber die Anzahl der Primzahlen unter einer gegebe-nen Gr ¨ osse191
12.3 J. Hadamard (1896) Sur la distribution des z ´ eros de la fonction (s) et ses cons ´ equences arithm ´ etiques207
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 ´ ee230
12.5 G. H. Hardy (1914) Sur les z ´ eros de la fonction (s) de Riemann304
12.6 G. H. Hardy (1915) Prime Numbers308
12.7 G. H. Hardy and J. E. Littlewood (1915) New Proofs of the Prime- Number Theorem and Simi-lar Theorems315
12.8 A. Weil (1941) On the Riemann hypothesis in Function-Fields321
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
References491
Index509
Index509