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Summary: Multiple zeta values at non-positive integers
Shigeki Akiyama and Yoshio Tanigawa
Abstract. Values of Euler-Zagier's multiple zeta function at non-positive integers
are studied, especially at (0; 0; : : : ; n) and ( n; 0; : : : ; 0). Further we prove a
symmetric formula among values at non-positive integers.
Key words: Multiple zeta function, Bernoulli numbers, Stirling numbers
1991 Mathematics Subject Classi cation: Primary 11M41, Secondary 11B68
1 Introduction.
One of remarkable properties of the Riemann zeta-function (s) is that
(2n) = ( 1) n+1 2 2n 1 B 2n
(2n)! 2n
for positive integers n, due to L. Euler. Here Bm denote the Bernoulli num-
bers de ned by x
e x 1
=
1
X
m=0
Bm
m! x m . Further he observed, by a farsighted
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