On Dirichlet series associated with polynomials

Carletti, E.; Monti Bragadin, G.

doi:10.1090/S0002-9939-1994-1179586-5

On Dirichlet series associated with polynomials
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by E. Carletti and G. Monti Bragadin PDF

Proc. Amer. Math. Soc. 121 (1994), 33-37 Request permission

Abstract:

Let $P(X)$ be a polynomial of degree N with complex coefficients and ${d_1},{d_2}$ two complex numbers with real part greater then $-1$. Consider the Dirichlet series associated with the triple $(P(X),{d_1},{d_2})$ \[ L(s) = \sum \limits _{n = 1}^\infty {\frac {{P(n)}}{{{{(n + {d_1})}^s}{{(n + {d_2})}^s}}}.} \] In this paper we get an explicit formula for $L(s)$ in terms of special functions which gives meromorphic continuation of $L(s)$ with at most simple poles at $s = (N + 1 - k)/2,k = 0,1, \ldots$ Finally we apply our explicit formula to Minakshisundaram’s zeta function of the three-dimensional sphere.

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Table of integrals, series and products

Eine Formel für den Logarithmus transzendenter Funktionen von endlichem Geschlecht

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