Springer Monographs in Mathematics

Bernoulli Numbers and Zeta Functions

Authors: Arakawa, Tsuneo, Ibukiyama, Tomoyoshi, Kaneko, Masanobu

  • Enables readers to begin reading without any prerequisite and smoothly guides them to more advanced topics in number theory
  • Provides repeated treatment, from different viewpoints, of both easy and advanced subjects related to Bernoulli numbers and zeta functions
  • Includes topics such as values of zeta functions, class numbers, exponential sums, Hurwitz numbers, multiple zeta functions, and poly-Bernoulli numbers
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About this book

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 doub

le 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.

About the authors

(late) Tsuneo Arakawa

Tomoyoshi Ibukiyama
Professor
Department of Mathematics
Graduate School of Science
Osaka University
Machikaneyama 1-1 Toyonaka, Osaka, 560-0043 Japan

Masanobu Kaneko
Professor
Faculty of Mathematics
Kyushu University
Motooka 744, Nishi-ku, Fukuoka, 819-0395, Japan

Reviews

“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)


Table of contents (14 chapters)

  • Bernoulli Numbers

    Ibukiyama, Tomoyoshi (et al.)

    Pages 1-24

  • Stirling Numbers and Bernoulli Numbers

    Ibukiyama, Tomoyoshi (et al.)

    Pages 25-39

  • Theorem of Clausen and von Staudt, and Kummer’s Congruence

    Ibukiyama, Tomoyoshi (et al.)

    Pages 41-49

  • Generalized Bernoulli Numbers

    Ibukiyama, Tomoyoshi (et al.)

    Pages 51-63

  • The Euler–Maclaurin Summation Formula and the Riemann Zeta Function

    Ibukiyama, Tomoyoshi (et al.)

    Pages 65-74

Buy this book

eBook $79.99
price for USA (gross)
  • ISBN 978-4-431-54919-2
  • Digitally watermarked, DRM-free
  • Included format: EPUB, PDF
  • ebooks can be used on all reading devices
  • Immediate eBook download after purchase
Hardcover $109.00
price for USA
  • ISBN 978-4-431-54918-5
  • Free shipping for individuals worldwide
  • Usually dispatched within 3 to 5 business days.
Softcover $109.00
price for USA
  • ISBN 978-4-431-56383-9
  • Free shipping for individuals worldwide
  • Usually dispatched within 3 to 5 business days.
Rent the ebook  
  • Rental duration: 1 or 6 month
  • low-cost access
  • online reader with highlighting and note-making option
  • can be used across all devices
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Bibliographic Information

Bibliographic Information
Book Title
Bernoulli Numbers and Zeta Functions
Authors
Series Title
Springer Monographs in Mathematics
Copyright
2014
Publisher
Springer Japan
Copyright Holder
Springer Japan
eBook ISBN
978-4-431-54919-2
DOI
10.1007/978-4-431-54919-2
Hardcover ISBN
978-4-431-54918-5
Softcover ISBN
978-4-431-56383-9
Series ISSN
1439-7382
Edition Number
1
Number of Pages
XI, 274
Number of Illustrations and Tables
4 b/w illustrations, 1 illustrations in colour
Topics