András Biró: Local square mean in the hyperbolic circle problem

Abstract:

Let \(\Gamma \subseteq \mathrm{PSL}_2(\mathbf{R})\) 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 \(\exp(2R/3)\) is known, and this has not been improved for any group. Petridis and Risager proved that in the special case \(\Gamma = \mathrm{PSL}_2(\mathbf{Z})\) taking \(z=w\) and averaging over \(z\) locally the error term can be improved. We show such an improvement for the local \(L^2\)-norm of the error term. Our estimate is better than the pointwise bound \(\exp(2R/3)\) but weaker than the bound of Petridis and Risager for the local average.