Local average of the hyperbolic circle problem for Fuchsian groups

TL;DR

This paper improves the error term in the hyperbolic circle problem for general Fuchsian groups by averaging over points, extending previous results that were limited to specific groups like PSL(2,Z).

Contribution

It generalizes the improved error estimate for the hyperbolic circle problem from PSL(2,Z) to all finite volume Fuchsian groups using a new version of the Selberg trace formula.

Findings

Error term improved to e^{(5/8 + ε) R} for general groups

Extension of averaging technique to broader class of Fuchsian groups

Utilization of generalized Selberg trace formula for analysis

Abstract

Let $\Gamma\subseteq PSL(2,{\bf R})$ be a finite volume Fuchsian group. The hyperbolic circle problem is the estimation of the number of elements of the $\Gamma$-orbit of $z$ in a hyperbolic circle around $w$ of radius $R$, where $z$ and $w$ are given points of the upper half plane and $R$ is a large number. An estimate with error term $e^{{2\over 3}R}$ is known, and this has not been improved for any group. Recently Risager and Petridis proved that in the special case $\Gamma =PSL(2,{\bf Z})$ taking $z=w$ and averaging over $z$ in a certain way the error term can be improved to $e^{\left({7\over {12}}+\epsilon\right)R}$. Here we show such an improvement for a general $\Gamma$, our error term is $e^{\left({5\over 8}+\epsilon\right)R}$ (which is better that $e^{{2\over 3}R}$ but weaker than the estimate of Risager and Petridis in the case $\Gamma =PSL(2,{\bf Z})$). Our main tool is our generalization of the Selberg trace formula proved earlier.