LOCAL AVERAGE OF THE HYPERBOLIC CIRCLE PROBLEM FOR FUCHSIAN GROUPS

Let $\unicode[STIX]{x1D6E4}\subseteq \operatorname{PSL}(2,\mathbf{R})$ be a finite-volume Fuchsian group. The hyperbolic circle problem is the estimation of the number of elements of the $\unicode[STIX]{x1D6E4}$ -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 $\text{e}^{(2/3)R}$ is known, and this has not been improved for any group. Recently, Risager and Petridis proved that in the special case $\unicode[STIX]{x1D6E4}=\operatorname{PSL}(2,\mathbf{Z})$ taking $z=w$ and averaging over $z$ in a certain way the error term can be improved to $\text{e}^{(7/12+\unicode[STIX]{x1D716})R}$ . Here we show such an improvement for a general $\unicode[STIX]{x1D6E4}$ ; our error term is $\text{e}^{(5/8+\unicode[STIX]{x1D716})R}$ (which is better than $\text{e}^{(2/3)R}$ but weaker than the estimate of Risager and Petridis in the case $\unicode[STIX]{x1D6E4}=\operatorname{PSL}(2,\mathbf{Z})$ ). Our main tool is our generalization of the Selberg trace formula proved earlier.