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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

The special values at negative integers of Dirichlet series associated with polynomials of several variables


Author: Min King Eie
Journal: Proc. Amer. Math. Soc. 119 (1993), 51-61
MSC: Primary 11M41; Secondary 11B68
MathSciNet review: 1148022
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Abstract: Let $ P(X)$ be a product of $ k$ linear forms in $ r$ variables $ {X_1}, \ldots ,{X_r}$ as given by

\begin{displaymath}\begin{array}{*{20}{c}} {P({X_1}, \ldots ,{X_r}) = \prod\limi... ...sum\limits_{i = 1}^r {{a_{ji}}} } \right) > 0.} \\ \end{array} \end{displaymath}

Suppose that $ \beta = ({\beta _1}, \ldots ,{\beta _r})$ is an $ r$-tuple of nonnegative integers. Consider the zeta function

$\displaystyle Z(P,\beta )(s) = \sum\limits_{{n_{1 = 1}}}^\infty { \cdots \sum\l... ...})}^{ - s}},\qquad \operatorname{Re} s > \frac{{r + \vert\beta \vert}} {k}} } ,$

where $ \vert\beta \vert = {\beta _1} + \cdots + {\beta _r}$. $ Z(P,\beta )(s)$ has an analytic continuation in the whole complex plane and it is regular at $ s = 0,\; - 1,\; - 2, \ldots , - m, \ldots $. In this paper, we shall compute the explicit values of $ Z(P,\beta )(s)$ at $ s = 0,\; - 1,\; - 2, \ldots ,\; - m, \ldots $ and express them in terms of finite sums of polynomials in Bernoulli numbers.

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1993-1148022-6
PII: S 0002-9939(1993)1148022-6
Article copyright: © Copyright 1993 American Mathematical Society