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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 de ned and their
properties are investigated.
1. Introduction
The multiple zeta values due to D. Zagier are de ned by
k (s 1 ; s 2 ; : : : ; s k ) =
X
0
1
n s1
1 n s2
2 : : : n s k
k
with positive integers s i (i = 1; 2; : : : ; k) and s k 2. These values have a
certain connection with topology and physics, and algebraic relations among
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