Springer Bernoulli Numbers and Zeta Functions (Springer Monographs in Mathematics)

Two major subjects are treated in this book. The main one is the theory of Bernoulli numbers and the other is the theory of zeta functions. Historically, Bernoulli numbers were introduced to give formulas for the sums of powers of consecutive integers. The real reason that they are indispensable for number theory, however, lies in the fact that special values of the Riemann zeta function can be written by using Bernoulli numbers. This leads to more advanced topics, a number of which are treated in this book: Historical remarks on Bernoulli numbers and the formula for the sum of powers of consecutive integers; a formula for Bernoulli numbers by Stirling numbers; the Clausen–von Staudt theorem on the denominators of Bernoulli numbers; Kummer's congruence between Bernoulli numbers and a related theory of p-adic measures; the Euler–Maclaurin summation formula; the functional equation of the Riemann zeta function and the Dirichlet L functions, and their special values at suitable integers; various formulas of exponential sums expressed by generalized Bernoulli numbers; the relation between ideal classes of orders of quadratic fields and equivalence classes of binary quadratic forms; class number formula for positive definite binary quadratic forms; congruences between some class numbers and Bernoulli numbers; simple zeta functions of prehomogeneous vector spaces; Hurwitz numbers; Barnes multiple zeta functions and their special values; the functional equation of the double zeta functions; and poly-Bernoulli numbers. An appendix by Don Zagier on curious and exotic identities for Bernoulli numbers is also supplied. This book will be enjoyable both for amateurs and for professional researchers. Because the logical relations between the chapters are loosely connected, readers can start with any chapter depending on their interests. The expositions of the topics are not always typical, and some parts are completely new. Review “The book touches on all of the well-known classical results related to Bernoulli numbers and zeta functions … . The book will offer something to readers at all levels of expertise, from the student of number theory looking for interesting topics to delve into, to researchers looking for an overview of various results, in each case pointing the way to further study.” (Luis Manuel Navas Vicente, Mathematical Reviews, October, 2015) “This book … is perhaps the first full-length treatment of these fascinating numbers―certainly the first modern one. … the book has an interdisciplinary character, offering thorough treatments of the Bernoulli numbers from the optics of the history of mathematics, combinatorics, analytic number theory, and algebraic number theory … . Summing Up: Highly recommended. Upper-division undergraduates and above.” (D. V. Feldman, Choice, Vol. 52 (10), June, 2015) “The present book contains some specific material reflecting the research interests of the authors. … The monograph is a useful addition to the library of every researcher working on special numbers and special functions.” (Khristo N. Boyadzhiev, zbMATH 1312.11015, 2015) “The book under review is about Bernoulli numbers and zeta functions. … The main audience for the book are researchers and students studying Bernoulli numbers and related topics. The text of the book is very fluent. Concepts and proofs are introduced in detail, and it is easy to follow for reader. There are some exercises, so the book can be used in a graduate course as well.” (Mehdi Hassani, MAA Reviews, December, 2014) From the Back Cover Two major subjects are treated in this book. The main one is the theory of Bernoulli numbers and the other is the theory of zeta functions. Historically, Bernoulli numbers were introduced to give formulas for the sums of powers of consecutive integers. The real reason that they are indispensable for number theory, however, lies in the fact that special values of the Riemann zeta func