# Spectral Exponential Sums on Hyperbolic Surfaces

## TL;DR

This paper derives an asymptotic formula for spectral exponential sums over Laplacian eigenvalues on hyperbolic surfaces, revealing how the sum's behavior depends on the nature of the Fuchsian group involved.

## Contribution

It introduces a new asymptotic formula connecting spectral sums with oscillatory components, von Mangoldt-like functions, and Selberg zeta functions for Maass cusp forms.

## Key findings

- Asymptotic behavior varies with the type of Fuchsian group
- Established connections between spectral sums and Selberg zeta functions
- Provided explicit formulas involving oscillatory components

## Abstract

We study an exponential sum over Laplacian eigenvalues $\lambda_{j} = 1/4+t_{j}^{2}$ with $t_{j} \leqslant T$ 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.