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Bernoulli numbers and zeta functions. With an appendix by Don Zagier. (English) Zbl 1312.11015

Springer Monographs in Mathematics. Tokyo: Springer (ISBN 978-4-431-54918-5/hbk; 978-4-431-54919-2/ebook). xi, 274 p. (2014).
The special numbers and special functions are experiencing increased popularity in the last 10–15 years and several books involving these topics have already been published. The present book contains some specific material reflecting the research interests of the authors.
The first two chapters provide a smooth and competent introduction to the Bernoulli and Stirling number, and chapter 4 deals with generalized Bernoulli numbers. A recent extension, the poly-Bernoulli numbers, appears in chapter 14. The authors describe two classical applications of Bernoulli numbers – the connection to the Riemann zeta function and the Euler-Maclaurin summation formula.
What distinguishes this book is the vast material on Bernoulli numbers and character sums, quadratic forms, congruences between Bernoulli numbers and class numbers, \(p\)-adic measure, and Kummer’s congruence.
The book has also a short chapter on Hurwitz numbers and a chapter on the Barnes multiple zeta function. A valuable feature is the Appendix by Don Zagier on “Curious and Exotic Identities of Bernoulli Numbers”. This includes Miki’s identity and related results.
The monograph is a useful addition to the library of every researcher working on special numbers and special functions.

MSC:

11B68 Bernoulli and Euler numbers and polynomials
11M06 \(\zeta (s)\) and \(L(s, \chi)\)
11-02 Research exposition (monographs, survey articles) pertaining to number theory
11M32 Multiple Dirichlet series and zeta functions and multizeta values
11B73 Bell and Stirling numbers
11S85 Other nonanalytic theory
33B99 Elementary classical functions
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