Analytic continuation of multiple zeta functions

Abstract

In this paper we shall define the analytic continuation of the multiple (Euler-Riemann-Zagier) zeta functions of depth d d : \[ ζ ( s 1 , … , s d ) := ∑ 0 > n 1 > n 2 > ⋯ > n d 1 n 1 s 1 n 2 s 2 ⋯ n d s d , \zeta (s_1,\dots ,s_d):= \sum _{0>n_1 > n_2>\cdots >n_d} \frac {1}{n_1^{s_1}n_2^{s_2}\cdots n_d^{s_d}}, \] where Re ⁡ ( s d ) > 1 \operatorname {Re}(s_d)>1 and ∑ j = 1 d Re ⁡ ( s j ) > d \sum _{j=1}^d\operatorname {Re}(s_j)>d . We shall also study their behavior near the poles and pose some open problems concerning their zeros and functional equations at the end.