Local square mean in the hyperbolic circle problem

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. Petridis and Risager proved that in the special case $\Gamma =PSL_2({\bf Z})$ taking $z=w$ and averaging over $z$ locally the error term can be improved to $e^{\left({7\over {12}}+\epsilon\right)R}$. Here we show such an improvement for the local $L^2$-norm of the error term. Our estimate is $e^{\left({9\over {14}}+\epsilon\right)R}$, which is better than the pointwise bound $e^{{2\over 3}R}$ but weaker than the bound of Petridis and Risager for the local average.