Analytic continuation of multiple zeta functions
Open Access
- 5 August 1999
- journal article
- Published by American Mathematical Society (AMS) in Proceedings of the American Mathematical Society
- Vol. 128 (5) , 1275-1283
- https://doi.org/10.1090/s0002-9939-99-05398-8
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.This publication has 5 references indexed in Scilit:
- Multiple Zeta Functions: An ExamplePublished by Mathematical Society of Japan (Project Euclid) ,2018
- Explicit evaluation of Euler sumsProceedings of the Edinburgh Mathematical Society, 1995
- Kontsevich's integral for the Homfly polynomial and relations between values of multiple zeta functionsTopology and its Applications, 1995
- Dirichlet series related to the Riemann zeta functionJournal of Number Theory, 1984
- Values of zeta-functions at non-negative integersLecture Notes in Mathematics, 1984