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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
DOI: https://doi.org/10.1090/S0002-9939-1994-1179586-5
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 $(P(X),{d_1},{d_2})$ \[ L(s) = \sum \limits _{n = 1}^\infty {\frac {{P(n)}}{{{{(n + {d_1})}^s}{{(n + {d_2})}^s}}}.} \] 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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Article copyright: © Copyright 1994 American Mathematical Society