Borwein / Choi / Rooney The Riemann Hypothesis

A Resource for the Afficionado and Virtuoso Alike
1. Auflage 2007
ISBN: 978-0-387-72126-2
Verlag: Springer
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 533 Seiten

Reihe: CMS Books in Mathematics

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.

Autoren/Hrsg.

Borwein, Peter

Choi, Stephen

Rooney, Brendan

Weirathmueller, Andrea

Inhaltsverzeichnis

1;Preface;7
2;Contents;9
3;Notation;12
4;Part I Introduction to the Riemann Hypothesis;14
4.1;1 Why This Book;15
4.1.1;1.1 The Holy Grail;15
4.1.2;1.2 Riemann’s Zeta and Liouville’s Lambda;17
4.1.3;1.3 The Prime Number Theorem;19
4.2;2 Analytic Preliminaries;21
4.2.1;2.1 The Riemann Zeta Function;21
4.2.2;2.2 Zero-free Region;28
4.2.3;2.3 Counting the Zeros of (s);30
4.2.4;2.4 Hardy’s Theorem;36
4.3;3 Algorithms for Calculating (s);40
4.3.1;3.1 Euler–MacLaurin Summation;40
4.3.2;3.2 Backlund;41
4.3.3;3.3 Hardy’s Function;42
4.3.4;3.4 The Riemann–Siegel Formula;43
4.3.5;3.5 Gram’s Law;44
4.3.6;3.6 Turing;45
4.3.7;3.7 The Odlyzko–Sch¨ onhage Algorithm;46
4.3.8;3.8 A Simple Algorithm for the Zeta Function;46
4.3.9;3.9 Further Reading;47
4.4;4 Empirical Evidence;48
4.4.1;4.1 Verification in an Interval;48
4.4.2;4.2 A Brief History of Computational Evidence;50
4.4.3;4.3 The Riemann Hypothesis and Random Matrices;51
4.4.4;4.4 The Skewes Number;54
4.5;5 Equivalent Statements;56
4.5.1;5.1 Number-Theoretic Equivalences;56
4.5.2;5.2 Analytic Equivalences;60
4.5.3;5.3 Other Equivalences;63
4.6;6 Extensions of the Riemann Hypothesis;66
4.6.1;6.1 The Riemann Hypothesis;66
4.6.2;6.2 The Generalized Riemann Hypothesis;67
4.6.3;6.3 The Extended Riemann Hypothesis;68
4.6.4;6.4 An Equivalent Extended Riemann Hypothesis;68
4.6.5;6.5 Another Extended Riemann Hypothesis;69
4.6.6;6.6 The Grand Riemann Hypothesis;69
4.7;7 Assuming the Riemann Hypothesis and Its Extensions . . .;72
4.7.1;7.1 Another Proof of The Prime Number Theorem;72
4.7.2;7.2 Goldbach’s Conjecture;73
4.7.3;7.3 More Goldbach;73
4.7.4;7.4 Primes in a Given Interval;74
4.7.5;7.5 The Least Prime in Arithmetic Progressions;74
4.7.6;7.6 Primality Testing;74
4.7.7;7.7 Artin’s Primitive Root Conjecture;75
4.7.8;7.8 Bounds on Dirichlet L-Series;75
4.7.9;7.9 The Lindel¨ of Hypothesis;76
4.7.10;7.10 Titchmarsh’s S( T ) Function;76
4.7.11;7.11 Mean Values of (s);77
4.8;8 Failed Attempts at Proof;79
4.8.1;8.1 Stieltjes and Mertens’ Conjecture;79
4.8.2;8.2 Hans Rademacher and False Hopes;80
4.8.3;8.3 Tur´ an’s Condition;81
4.8.4;8.4 Louis de Branges’s Approach;81
4.8.5;8.5 No Really Good Idea;82
4.9;9 Formulas;83
4.10;10 Timeline;90
5;Part II Original Papers;100
5.1;11 Expert Witnesses;101
5.1.1;11.1 E. Bombieri (2000–2001) Problems of the Millennium: The Riemann Hypothesis;102
5.1.2;11.2 P. Sarnak (2004) Problems of the Millennium: The Riemann Hypothesis;114
5.1.3;11.3 J. B. Conrey (2003) The Riemann Hypothesis;124
5.1.4;11.4 A. Ivi ´ c (2003) On Some Reasons for Doubting the Riemann Hypothesis;138
5.2;12 The Experts Speak for Themselves;169
5.2.1;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
5.2.2;12.2 B. Riemann (1859) Ueber die Anzahl der Primzahlen unter einer gegebe-nen Gr ¨ osse;191
5.2.3;12.3 J. Hadamard (1896) Sur la distribution des z ´ eros de la fonction (s) et ses cons ´ equences arithm ´ etiques;207
5.2.4;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
5.2.5;12.5 G. H. Hardy (1914) Sur les z ´ eros de la fonction (s) de Riemann;304
5.2.6;12.6 G. H. Hardy (1915) Prime Numbers;308
5.2.7;12.7 G. H. Hardy and J. E. Littlewood (1915) New Proofs of the Prime- Number Theorem and Simi-lar Theorems;315
5.2.8;12.8 A. Weil (1941) On the Riemann hypothesis in Function-Fields;321
5.2.9;12.9 P. Turan (1948);325
5.2.10;12.10 A. Selberg (1949);361
5.2.11;12.11 P. Erdös (1949);371
5.2.12;12.12 S. Skewes (1955);383
5.2.13;12.13 C. B. Haselgrove (1958);407
5.2.14;12.14 H. Montgomery (1973);413
5.2.15;12.15 D. J. Newman (1980);427
5.2.16;12.16 J. Korevaar (1982);432
5.2.17;12.17 H. Daboussi (1984);441
5.2.18;12.18 A. Hildebrand (1986);446
5.2.19;12.19 D. Goldston and H. Montgomery (1987);455
5.2.20;12.20 M. Agrawal, N. Kayal, and N. Saxena (2004);477
6;References;491
7;Index;509
8;Index;509