Local square mean in the hyperbolic circle problem

Let Γ⊆ 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 e^2 3R 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 to e^(7 12+ε)R. Here we show such an improvement for the local L²-norm of the error term. Our estimate is e^(9 14+ε)R, which is better than the pointwise bound e^2 3R but weaker than the bound of Petridis and Risager for the local average.