Local square mean in the hyperbolic circle problem

Let  PSL 2 ‫/ޒ.‬ 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 2R=3 is known, and this has not been improved for any group.Petridis and Risager proved that in the special case D PSL 2 ‫/ޚ.‬ taking z D w and averaging over z locally the error term can be improved to e .7=12C/R .Here we show such an improvement for the local L 2 -norm of the error term.Our estimate is e .9=14C/R , which is better than the pointwise bound e 2R=3 but weaker than the bound of Petridis and Risager for the local average.Research partially supported by the NKFIH (National Research, Development and Innovation Office) grants K135885 and K143876, and by the Rényi Intézet Lendület Automorphic Research Group.MSC2020: 11F72.