Petridis, YN;
Risager, MS;
(2017)
Averaging over Heegner points in the hyperbolic circle problem.
International Mathematics Research Notices
, 2018
(16)
pp. 4942-4968.
10.1093/imrn/rnx026.
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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.
| Type: | Article |
|---|---|
| Title: | Averaging over Heegner points in the hyperbolic circle problem |
| Open access status: | An open access version is available from UCL Discovery |
| DOI: | 10.1093/imrn/rnx026 |
| Publisher version: | https://dx.doi.org/10.1093/imrn/rnx026 |
| Language: | English |
| Additional information: | This version is the author accepted manuscript. For information on re-use, please refer to the publisher’s terms and conditions. |
| Keywords: | Mathematics, Number Theory, 11F72 |
| UCL classification: | UCL UCL > Provost and Vice Provost Offices UCL > Provost and Vice Provost Offices > UCL BEAMS UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences > Dept of Mathematics |
| URI: | https://discovery.ucl.ac.uk/id/eprint/1570527 |
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