As its title indicates, the book under review is about Bernoulli numbers and zeta functions. Several properties and generalizations of Bernoulli numbers, and various relations to the Riemann zeta function and some other zeta functions are discussed.
In Chapter 1, authors review the history of Bernoulli numbers and study two equivalent definitions and some basic properties of these numbers. More precisely, they describe the role of Bernoulli numbers in studying the sum of powers of consecutive natural numbers. In Chapter 2, authors study Stirling numbers and their relations with Bernoulli numbers. Chapter 3 is concerned with the theorem of Clausen and von Staudt giving the fractional part of a Bernoulli number as a finite sum of the reciprocals of certain prime numbers. This chapter also includes and Kummer's congruences.
Chapter 4 starts by introducing Dirichlet characters and gives a generalization of Bernoulli numbers and Bernoulli functions related by Dirichlet characters. In this chapter, authors give a different definition of Bernoulli functions suggested by Don Zagier. In Chapter 5 authors survey the Euler-Maclaurin summation formula and the values of the Riemann zeta function at integer arguments.
In Chapters 6 and 7 authors study relations between Bernoulli numbers and quadratic forms and prove a congruence relation between Bernoulli numbers and class numbers of imaginary quadratic fields. The authors provide several identities between Bernoulli numbers and the roots of unity in Chapter 8.
As a continuation of Chapters 4 and 5, in Chapter 9 the authors study some properties of Hurwitz zeta functions and Dirichlet L-functions, calculating their special values at negative integers, mainly in terms of Bernoulli and generalized Bernoulli numbers. In Chapter 10, as an application of quadratic forms and quadratic fields, they give an explicit formula for some simple zeta functions related to so-called prehomogeneous vector spaces. They also prove a class number formula for imaginary quadratic fields.
In Chapter 11 authors explain the connection between Bernoulli numbers and p-adic integrals. They define and study a specific measure called the Bernoulli measure and use it to prove Kummer’s congruences. In Chapter 12, authors study Hurwitz’s generalization of the Bernoulli numbers, known as the Hurwitz numbers. In Chapter 13 they introduce Barnes’s multiple zeta function, which is a natural generalization of the Hurwitz zeta function, give an analytic continuation, and then express their special values at negative integers by using Bernoulli polynomials. Finally, in Chapter 14 they define and study a generalization of Bernoulli numbers referred to as poly-Bernoulli numbers, which is a different generalization than the generalized Bernoulli numbers introduced in Chapter 4.
The book ends with an appendix by Don Zagier, which can be read independently of the main text, describing some of the less standard properties, mainly based of concepts related to generating 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 is a faculty member at the Department of Mathematics, Zanjan University, Iran. His fields of interest are Elementary, Analytic and Probabilistic Number Theory.