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arXiv · 2604.11205

Local square mean in the hyperbolic circle problem and sums of Sali\'e sums

András Biró
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Abstract

Let $\Gamma\subseteq PSL(2, \mathbb R)$ be a finite volume Fuchsian group. The hyperbolic circle problem is the estimation of the number of elements of the $\Gamma$-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 $e^{\frac 23R}$ is known, and this has not been improved for any group. Recently, taking $ z=w$ and considering $\Gamma = PSL(2, \mathbb Z)$, we have shown the estimate $ e^{\left(\frac 9{14}+\epsilon\right)R}$ for the local $L^2$-norm of the error term, which is better than the pointwise bound. Here we improve the exponent $\frac 9{14}$, conditionally on a twisted Linnik-Selberg-type conjecture for sums of Sali\'e sums.

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András Biró. 2026-04-13. Local square mean in the hyperbolic circle problem and sums of Sali\'e sums. https://arxiv.org/abs/2604.11205

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