Averaging over Heegner points in the hyperbolic circle problem

TL;DR

This paper improves the error estimates in the hyperbolic circle problem for the modular group by averaging over Heegner points, leveraging bounds on spectral sums and conjectures in number theory.

Contribution

It introduces a novel averaging technique over Heegner points to enhance error bounds in the hyperbolic circle problem, connecting spectral bounds with deep conjectures.

Findings

Improved error term estimates for large discriminant D.

Utilization of spectral exponential sum bounds.

Connections to the Lindelöf and sup-norm conjectures.

Abstract

For $\Gamma={\hbox{PSL}_2( {\mathbb Z})}$ the hyperbolic circle problem aims to estimate the number of elements of the orbit $\Gamma z$ inside the hyperbolic disc centered at $z$ with radius $\cosh^{-1}(X/2)$. We show that, by averaging over Heegner points $z$ of discriminant $D$, Selberg's error term estimate can be improved, if $D$ is large enough. The proof uses bounds on spectral exponential sums, and results towards the sup-norm conjecture of eigenfunctions, and the Lindel\"of conjecture for twists of the $L$-functions attached to Maa{\ss} cusp forms.