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ON ANALYTIC CONTINUATION OF MULTIPLE L-FUNCTIONS AND RELATED ZETA-FUNCTIONS
 

Summary: ON ANALYTIC CONTINUATION OF MULTIPLE L-FUNCTIONS
AND RELATED ZETA-FUNCTIONS
SHIGEKI AKIYAMA & HIDEAKI ISHIKAWA
1. Introduction
Analytic continuation of Euler-Zagier's multiple zeta function of two variables
was rst established by F.V.Atkinson [3] with an application to the mean value
problem of the Riemann zeta function. We can nd recent developments in [8],[7]
and [5]. From an analytic point of view, these results suggest broad applications
of multiple zeta functions. In [9] and [10], D.Zagier pointed out an interesting
interplay between positive integer values and other areas of mathematics, which in-
clude knot theory and mathematical physics. Many works had been done according
to his motivation but here we restrict our attention to the analytic continuation.
T.Arakawa and M.Kaneko [2] showed an analytic continuation with respect to the
last variable. To speak about the analytic continuation with respect to all variables,
we have to refer to J. Zhao [11] and S.Akiyama, S.Egami and Y.Tanigawa [1]. In
[11], an analytic continuation and the residue calculation were done by using the
theory of generalized functions in the sense of I.M. Gel'fand and G.E. Shilov. In [1],
they gave an analytic continuation by means of a simple application of the Euler-
Maclaurin formula. The advantage of this method is that it gives the complete
location of singularities. This work also includes some study on the values at non

  

Source: Akiyama, Shigeki - Department of Mathematics, Niigata University

 

Collections: Mathematics