András Biró: Local square mean in the hyperbolic circle problem

Speaker
András Biró: Renyi Institute, Budapest

Abstract
Let \(\Gamma\subseteq PSL2(R)\) be a finite volume Fuchsian group. The hyperbolic circle problem is the estimation of the number of elements of the \(Γ\)-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 \(Γ=PSL2(Z)\) 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.