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# On a Dirichlet Series Associated with a Polynomial

Minking Eie
Proceedings of the American Mathematical Society
Vol. 110, No. 3 (Nov., 1990), pp. 583-590
DOI: 10.2307/2047897
Stable URL: http://www.jstor.org/stable/2047897
Page Count: 8
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## Abstract

Let $P(x) = \prod^k_{j = 1}(x + \delta_j)$ be a polynomial with real coefficients and $\mathrm{Re} \delta_j > -1 (j = 1,\ldots, k)$. Define the zeta function ZP(s) associated with the polynomial P(x) as $Z_P(s) = \sum^\infty_{n = 1} \frac{1}{P(n)^s}, \mathrm{Re} s > 1/k.$ ZP(s) is holomorphic for $\mathrm{Re} s > 1/k$ and it has an analytic continuation in the whole complex s-plane with only possible simple poles at s = j/k (j = 1, 0, -1, -2, -3,...) other than nonpositive integers. In this paper, we shall obtain the explicit value of ZP(-m) for any nonnegative integer m, the asymptotic formula of ZP(s) at s = 1/k, the value Z'P(0) and its application to the determinants of elliptic operators.

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