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Analytic continuation of multiple zeta functions

Author: Jianqiang Zhao
Journal: Proc. Amer. Math. Soc. 128 (2000), 1275-1283
MSC (1991): Primary 11M99; Secondary 30D30, 30D10
Published electronically: August 5, 1999
MathSciNet review: 1670846
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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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Additional Information

Jianqiang Zhao
Affiliation: Department of Mathematics, Brown University, Providence, Rhode Island 02912
Address at time of publication: Department of Mathematics, University of Pennsylvania, Philadelphia, Pennsylvania 19104

Keywords: Analytic continuation, multiple zeta function, generalized function
Received by editor(s): June 21, 1998
Published electronically: August 5, 1999
Communicated by: Dennis A. Hejhal
Article copyright: © Copyright 2000 American Mathematical Society