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The Riemann hypothesis. A resource for the afficionado and virtuoso alike. (English) Zbl 1132.11047
CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC. New York, NY: Springer (ISBN 978-0-387-72125-5/hbk). xiv, 533 p. EUR 62.95/net; SFR 110.00; $ 79.95; £ 48.50 (2007).

This book is intended as a reference work on the Riemann Hypothesis (RH). It proves some of the basic theorems on the Riemann Zeta-function, and describes many more. These cover, in the first part of the book, analytic preliminaries, algorithms for calculating ζ(s), evidence for RH, statements equivalent to RH, generalizations of RH, deductions from RH, and attempts to prove RH. The first part of the book concludes with a formulary and a timeline.

The second, longer, part of the book contains reproductions of four survey articles, by Bombieri, Sarnak, Conrey and Ivić, and 20 significant original research papers.

This book will undoubtedly be extremely useful for anyone making a serious study of the zeta-function, and especially those with an interest in the historical development of the subject. The choice of material is good, and the discussion is helpful. Many of the papers reproduced are difficult to track down nowadays, and anyone working in the area will benefit from a study of them.

Overall this is a book which belongs on the shelves of anyone interested in the RH.

11M26Nonreal zeros of ζ(s) and L(s,χ); Riemann and other hypotheses
11-02Research monographs (number theory)
11-03Historical (number theory)
00B60Collections of reprinted articles