2024/03/24 by Biró, András · 3 citations
#FOS: Mathematics #Number Theory (math.NT)
paper · doi:10.48550/arxiv.2403.16113
Let Γ⊆ PSL2(\bf 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\over 3R is known, and this has not been improved for any group. Petridis and Risager proved that in the special case Γ=PSL2(\bf Z) taking z=w and averaging over z locally the error term can be improved to e^(7\over 12+ε)R. Here we show such an improvement for the local L2-norm of the error term. Our estimate is e^(9\over 14+ε)R, which is better than the pointwise bound e^2\over 3R but weaker than the bound of Petridis and Risager for the local average.