Spectral Exponential Sums on Hyperbolic Surfaces

Abstract

We study an exponential sum over Laplacian eigenvalues ${\lambda }_j=1/4+t_j^2$ with $t_j⩽T$ for Maass cusp forms on $\Gamma \backslash \mathbb{H}$, where $\Gamma$ is a cofinite Fuchsian group acting on the upper half-plane $\mathbb{H}$. The aim is to establish an asymptotic formula which expresses spectral exponential sums in terms of an oscillatory component, von Mangoldt-like functions and Selberg zeta functions. The behaviour is determined by whether $\Gamma$ is essentially cuspidal or not.

8 pages

References in corpus (1)

• Euler products of Selberg zeta functions in the critical strip

Cited by in corpus (2)

• The Prime Geodesic Theorem for ${\mathrm{PSL}}_2(\mathbb{Z}[i])$ and Spectral Exponential Sums
• The Prime Geodesic Theorem in Arithmetic Progressions