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Riemann Hypothesis A Resource for the Afficionado and Virtuoso Alike

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ISBN-10: 0387721258

ISBN-13: 9780387721255

Edition: 2008

Authors: Peter Borwein, Stephen Choi, Brendan Rooney, Andrea Weirathmueller

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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. This collection is not very large and encompasses only 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.
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Book details

Copyright year: 2008
Publisher: Springer
Publication date: 11/30/2007
Binding: Hardcover
Pages: 533
Size: 6.25" wide x 9.25" long x 1.25" tall
Weight: 1.936
Language: English

Introduction to the Riemann Hypothesis
Why This Book
The Holy Grail
Riemann's Zeta and Liouville's Lambda
The Prime Number Theorem
Analytic Preliminaries
The Riemann Zeta Function
Zero-free Region
Counting the Zeros of [zeta](s)
Hardy's Theorem
Algorithms for Calculating [zeta](s)
Euler-MacLaurin Summation
Hardy's Function
The Riemann-Siegel Formula
Gram's Law
The Odlyzko-Schonhage Algorithm
A Simple Algorithm for the Zeta Function
Further Reading
Empirical Evidence
Verification in an Interval
A Brief History of Computational Evidence
The Riemann Hypothesis and Random Matrices
The Skewes Number
Equivalent Statements
Number-Theoretic Equivalences
Analytic Equivalences
Other Equivalences
Extensions of the Riemann Hypothesis
The Riemann Hypothesis
The Generalized Riemann Hypothesis
The Extended Riemann Hypothesis
An Equivalent Extended Riemann Hypothesis
Another Extended Riemann Hypothesis
The Grand Riemann Hypothesis
Assuming the Riemann Hypothesis and Its Extensions
Another Proof of The Prime Number Theorem
Goldbach's Conjecture
More Goldbach
Primes in a Given Interval
The Least Prime in Arithmetic Progressions
Primality Testing
Artin's Primitive Root Conjecture
Bounds on Dirichlet L-Series
The Lindelof Hypothesis
Titchmarsh's S(T) Function
Mean Values of [zeta](s)
Failed Attempts at Proof
Stieltjes and Mertens' Conjecture
Hans Rademacher and False Hopes
Turan's Condition
Louis de Branges's Approach
No Really Good Idea
Original Papers
Expert Witnesses
E. Bombieri (2000-2001)
P. Sarnak (2004)
J. B. Conrey (2003)
A. Ivic (2003)
The Experts Speak for Themselves
P. L. Chebyshev (1852)
B. Riemann (1859)
J. Hadamard (1896)
C. de la Vallee Poussin (1899)
G. H. Hardy (1914)
G. H. Hardy (1915)
G. H. Hardy and J. E. Littlewood (1915)
A. Weil (1941)
P. Turan (1948)
A. Selberg (1949)
P. Erdos (1949)
S. Skewes (1955)
C. B. Haselgrove (1958)
H. Montgomery (1973)
D. J. Newman (1980)
J. Korevaar (1982)
H. Daboussi (1984)
A. Hildebrand (1986)
D. Goldston and H. Montgomery (1987)
M. Agrawal, N. Kayal, and N. Saxena (2004)