On the Analytic Continuation of Various Multiple Zeta-Functions
Kohji Matsumoto
Abstract
In this article we give a survey of the history of the problem of analytic continuation of multiple zeta-functions, and we prove some new results in this connection. We begin in Section 1 by describing the work of E. W. Barnes and H. Mellin at the turn of the 20th century. In Sections 2 and 3 we discuss the Euler sum and its multi-variable generalization, which recently have become again the subject of active research. In Section 4 we describe a new method of M. Katsurada which uses the classical Mellin-Barnes integral formula to establish the analytic continuation of the Euler sum. In the final two sections we present new results of the author, obtained by applying the Mellin-Barnes formula to more general multiple zeta-functions.