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ANALYTIC CONTINUATION OF MULTIPLE ZETA-FUNCTIONS AND THEIR VALUES
 

Summary: ANALYTIC CONTINUATION OF MULTIPLE
ZETA-FUNCTIONS AND THEIR VALUES
AT NON-POSITIVE INTEGERS
SHIGEKI AKIYAMA, SHIGEKI EGAMI AND YOSHIO TANIGAWA
Abstract. Analytic continuation of the multiple zeta-function is es-
tablished by a simple application of the Euler-Maclaurin summation for-
mula. Multiple zeta values at non-positive integers are defined and their
properties are investigated.
1. Introduction
The multiple zeta values due to D. Zagier are defined by
k(s1, s2, . . . , sk) =
0 1
ns1
1 ns2
2 . . . nsk
k
with positive integers si (i = 1, 2, . . . , k) and sk 2. These values have a
certain connection with topology and physics, and algebraic relations among
them are extensively studied (see [18], [19], [6], [7] and [14]). Recently,

  

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

 

Collections: Mathematics