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

 

On Dirichlet series associated with polynomials


Authors: E. Carletti and G. Monti Bragadin
Journal: Proc. Amer. Math. Soc. 121 (1994), 33-37
MSC: Primary 11M35; Secondary 11M41, 58G26
MathSciNet review: 1179586
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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})$

$\displaystyle 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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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1994-1179586-5
PII: S 0002-9939(1994)1179586-5
Article copyright: © Copyright 1994 American Mathematical Society