 
Summary: ON ANALYTIC CONTINUATION OF MULTIPLE LFUNCTIONS
AND RELATED ZETAFUNCTIONS
SHIGEKI AKIYAMA & HIDEAKI ISHIKAWA
1. Introduction
Analytic continuation of EulerZagier's multiple zeta function of two variables
was first established by F.V.Atkinson [3] with an application to the mean value
problem of the Riemann zeta function. We can find 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
