Bernoulli Numbers and Zeta Functions

  • Tsuneo Arakawa
  • Tomoyoshi Ibukiyama
  • Masanobu Kaneko

Part of the Springer Monographs in Mathematics book series (SMM)

Table of contents

  1. Front Matter
    Pages i-xi
  2. Tomoyoshi Ibukiyama, Masanobu Kaneko
    Pages 1-24
  3. Tomoyoshi Ibukiyama, Masanobu Kaneko
    Pages 25-39
  4. Tomoyoshi Ibukiyama, Masanobu Kaneko
    Pages 41-49
  5. Tomoyoshi Ibukiyama, Masanobu Kaneko
    Pages 51-63
  6. Tomoyoshi Ibukiyama, Masanobu Kaneko
    Pages 65-74
  7. Tomoyoshi Ibukiyama, Masanobu Kaneko
    Pages 75-93
  8. Tomoyoshi Ibukiyama, Masanobu Kaneko
    Pages 103-137
  9. Tomoyoshi Ibukiyama, Masanobu Kaneko
    Pages 139-153
  10. Tomoyoshi Ibukiyama, Masanobu Kaneko
    Pages 155-182
  11. Tomoyoshi Ibukiyama, Masanobu Kaneko
    Pages 183-201
  12. Tomoyoshi Ibukiyama, Masanobu Kaneko
    Pages 203-208
  13. Tomoyoshi Ibukiyama, Masanobu Kaneko
    Pages 209-222
  14. Tomoyoshi Ibukiyama, Masanobu Kaneko
    Pages 223-238
  15. Back Matter
    Pages 239-274

About this book

Introduction

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.

Keywords

Bernoulli numbers and polynomials L-functions MSC; 11B68, 11B73, 11M06, 11L03, 11M06, 11M32, 11M35 Riemann zeta function Stirling numbers exponential sums

Authors and affiliations

  • Tsuneo Arakawa
    • 1
  • Tomoyoshi Ibukiyama
    • 2
  • Masanobu Kaneko
    • 3
  1. 1.Department MathematicsRikkyo UniversityTokyoJapan
  2. 2.Osaka UniversityOsakaJapan
  3. 3.Kyushu UniversityFukuokaJapan

Bibliographic information

  • DOI https://doi.org/10.1007/978-4-431-54919-2
  • Copyright Information Springer Japan 2014
  • Publisher Name Springer, Tokyo
  • eBook Packages Mathematics and Statistics
  • Print ISBN 978-4-431-54918-5
  • Online ISBN 978-4-431-54919-2
  • Series Print ISSN 1439-7382
  • Series Online ISSN 2196-9922
  • About this book