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On a Dirichlet series associated with a polynomial


Author: Min King Eie
Journal: Proc. Amer. Math. Soc. 110 (1990), 583-590
MSC: Primary 11M41; Secondary 11F66
DOI: https://doi.org/10.1090/S0002-9939-1990-1037206-0
MathSciNet review: 1037206
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Abstract: Let $ P(x) = \prod\nolimits_{j = 2}^k {(x + {\delta _j})} $ be a polynomial with real coefficients and $ \operatorname{Re} {\delta _j} > - 1(j = 1, \ldots ,k)$. Define the zeta function $ {Z_p}(s)$ associated with the polynomial $ P(x)$ as

$\displaystyle {Z_P}(s) = \sum\limits_{n = 1}^\infty {\frac{1}{{P{{(n)}^s}}}} ,\operatorname{Re} s > 1/k.$

$ Z_P(s)$ is holomorphic for $ \operatorname{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, \ldots )$ other than nonpositive integers.

In this paper, we shall obtain the explicit value of $ {Z_P}( - m)$ for any non-negative integer $ m$, the asymptotic formula of $ {Z_P}(s)$ at $ s = 1/k$, the value $ {Z'_P}(0)$ and its application to the determinants of elliptic operators.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1990-1037206-0
Article copyright: © Copyright 1990 American Mathematical Society

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