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arXiv · 1610.09393

Averaging over Heegner points in the hyperbolic circle problem

Yiannis N. Petridis·Morten S. Risager
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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.

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Yiannis N. Petridis, Morten S. Risager. 2016-10-28. Averaging over Heegner points in the hyperbolic circle problem. https://arxiv.org/abs/1610.09393

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