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OPPOSITE-SIGN KLOOSTERMAN SUM ZETA FUNCTION

Published online by Cambridge University Press:  28 January 2016

Eren Mehmet Kıral
Affiliation:
Department of Mathematics, Texas A&M University, Mailstop 3368, College Station, TX 77843, U.S.A. email ekiral@tamu.edu
Corresponding
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Abstract

We study the meromorphic continuation and the spectral expansion of the opposite-sign Kloosterman sum zeta function

$$\begin{eqnarray}\displaystyle (2{\it\pi}\sqrt{mn})^{2s-1}\mathop{\sum }_{\ell =1}^{\infty }\frac{S(m,-n,\ell )}{\ell ^{2s}} & & \displaystyle \nonumber\end{eqnarray}$$
for $m,n$ positive integers, to all $s\in \mathbb{C}$ . There are poles of the function corresponding to zeros of the Riemann zeta function and the spectral parameters of Maass forms. The analytic properties of this function are rather delicate. It turns out that the spectral expansion of the zeta function converges only in a left half-plane, disjoint from the region of absolute convergence of the Dirichlet series, even though they both are analytic expressions of the same meromorphic function on the entire complex plane.

Type
Research Article
Copyright
Copyright © University College London 2016 

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References

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