On a Dirichlet series associated with a polynomial

Eie, Min King

doi:10.1090/S0002-9939-1990-1037206-0

On a Dirichlet series associated with a polynomial
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by Min King Eie PDF

Proc. Amer. Math. Soc. 110 (1990), 583-590 Request permission

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 \[ {Z_P}(s) = \sum \limits _{n = 1}^\infty {\frac {1}{{P{{(n)}^s}}}} ,\operatorname {Re} s > 1/k.\] $Z_P(s)$ 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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