Local average of the hyperbolic circle problem for Fuchsian groups

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.