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Analytic continuation of multiple zeta functions

Author(s): Jianqiang Zhao
Journal: Proc. Amer. Math. Soc. 128 (2000), 1275-1283.
MSC (1991): Primary 11M99; Secondary 30D30, 30D10
Posted: 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 :

$\begin{displaymath}\zeta(s_1,\dots,s_d):= \sum _{0<n_1 < n_2<\cdots<n_d} \frac{1}{n_1^{s_1}n_2^{s_2}\cdots n_d^{s_d}},\end{displaymath}$

where $\re(s_d)>1$ and $\sum _{j=1}^d\re(s_j)>d$ . 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
Email: jzhao@math.brown.edu

DOI: 10.1090/S0002-9939-99-05398-8
PII: S 0002-9939(99)05398-8
Keywords: Analytic continuation, multiple zeta function, generalized function
Received by editor(s): June 21, 1998
Posted: August 5, 1999
Communicated by: Dennis A. Hejhal
Copyright of article: Copyright 2000, American Mathematical Society

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