 
Summary: ANALYTIC CONTINUATION OF MULTIPLE
ZETAFUNCTIONS AND THEIR VALUES
AT NONPOSITIVE INTEGERS
SHIGEKI AKIYAMA, SHIGEKI EGAMI AND YOSHIO TANIGAWA
Abstract. Analytic continuation of the multiple zetafunction is es
tablished by a simple application of the EulerMaclaurin summation for
mula. Multiple zeta values at nonpositive 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,
