Opposite Sign Kloosterman Sum Zeta Function

TL;DR

This paper investigates the complex analytic properties and spectral expansion of the opposite sign Kloosterman sum zeta function, revealing intricate pole structures related to the Riemann zeta zeros and Maass form spectral parameters.

Contribution

It provides the meromorphic continuation and spectral expansion analysis of the opposite sign Kloosterman sum zeta function across the entire complex plane, highlighting convergence and pole structure.

Findings

Spectral expansion converges only in a left half-plane.

Poles correspond to zeros of the Riemann zeta function.

Analytic properties are delicate and complex.

Abstract

We study the meromorphic continuation and the spectral expansion of the oppposite sign Kloosterman sum zeta function, $$(2\pi \sqrt{mn})^{2s-1}\sum_{\ell=1}^\infty \frac{S(m,-n,\ell)}{\ell^{2s}}$$ 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.