Topics in Number Theory (Monographs in Number Theory)

This is a first-ever textbook written in English about the theory of modular forms and Jacobi forms of several variables. It contains the classical theory as well as a new theory on Jacobi forms over Cayley numbers developed by the author from 1990 to 2000. Applications to the classical Euler sums are of special interest to those who are eager to evaluate double Euler sums or more general multiple zeta values. The celebrated sum formula proved by Granville in 1997 is given in a more general form here.

Contents:Theory of Modular Forms of One Variable:; Group Action of the Modular Group; The Gamma and Zeta Functions; Zeta Functions of Modular Forms; Dimension Formulae; Bernoulli Identities and Applications; Euler Sums and Recent Development; Theory of Modular Forms of Several Variables:; Theory of Modular Forms of Several Variables; The Full Modular Group; The Fourier Coefficients of Eisenstein Series; Theory of Jacobi Forms; Hecke Operators and Jacobi Forms; Singular Modular Forms on the Exceptional Domain.