Analytic continuation of multiple zeta functions

Zhao, Jianqiang

doi:10.1090/S0002-9939-99-05398-8

Analytic continuation of multiple zeta functions
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by Jianqiang Zhao PDF

Proc. Amer. Math. Soc. 128 (2000), 1275-1283 Request permission

Abstract:

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

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