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On the analytic continuation of various multiple zeta-functions. (English) Zbl 1031.11051
Bennett, M. A. (ed.) et al., Number theory for the millennium II. Proceedings of the millennial conference on number theory, Urbana-Champaign, IL, USA, May 21-26, 2000. Natick, MA: A K Peters. 417-440 (2002).
The author gives an interesting survey of the history of the problem of analytic continuation of multiple zeta-functions and proves some new results in this connection. He begins by describing the work of E. W. Barnes and H. Mellin at the turn of the 20th century. Then he discusses the Euler sum and its multi-variable generalization. Further he describes a new method of M. Katsurada [see Collect. Math. 48, 137-153 (1997; Zbl 0891.11042) and Lith. Math. J. 38, 77-88 (1998; Zbl 0924.11073)] which uses the classical Mellin-Barnes integral formula to establish the analytic continuation of the Euler sum. Also in the paper new results of the author are presented that are obtained by applying the Mellin-Barnes formula to more general multiple zeta-functions.
For the entire collection see [Zbl 1002.00006].

##### MSC:
 11M41 Other Dirichlet series and zeta functions
##### Keywords:
multiple zeta-function; analytic continuation