Petridis, YN;
Risager, MS;
(2017)
Local average in hyperbolic lattice point counting, with an Appendix by Niko Laaksonen.
Mathematische Zeitschrift
, 285
(3-4)
pp. 1319-1344.
10.1007/s00209-016-1749-z.
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Abstract
The hyperbolic lattice point problem asks to estimate the size of the orbit Γz inside a hyperbolic disk of radius cosh−1(X/2) for Γ a discrete subgroup of PSL2(R). Selberg proved the estimate O(X2/3) for the error term for cofinite or cocompact groups. This has not been improved for any group and any center. In this paper local averaging over the center is investigated for PSL2(Z). The result is that the error term can be improved to O(X7/12+ε). The proof uses surprisingly strong input e.g. results on the quantum ergodicity of Maaß cusp forms and estimates on spectral exponential sums. We also prove omega results for this averaging, consistent with the conjectural best error bound O(X1/2+ε). In the appendix the relevant exponential sum over the spectral parameters is investigated.
| Type: | Article |
|---|---|
| Title: | Local average in hyperbolic lattice point counting, with an Appendix by Niko Laaksonen |
| Open access status: | An open access version is available from UCL Discovery |
| DOI: | 10.1007/s00209-016-1749-z |
| Publisher version: | http://dx.doi.org/10.1007/s00209-016-1749-z |
| 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. |
| 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/1521410 |
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