Local square mean in the hyperbolic circle problem

Let PSL₂ (R) be a finite volume Fuchsian group. The hyperbolic circle problem is the estimation of the number of elements of the -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 =PSL₂ (Z) taking z=w and averaging over z locally the error term can be improved to e^ ({7 {12}+) R}. Here we show such an improvement for the local L²-norm of the error term. Our estimate is e^ ({9 {14}+) R}, which is better than the pointwise bound e^{2 3R} but weaker than the bound of Petridis and Risager for the local average.