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Mathematische Zeitschrift

, Volume 285, Issue 3–4, pp 1319–1344 | Cite as

Local average in hyperbolic lattice point counting, with an Appendix by Niko Laaksonen

  • Yiannis N. Petridis
  • Morten S. Risager
Article

Abstract

The hyperbolic lattice point problem asks to estimate the size of the orbit \(\Gamma z\) inside a hyperbolic disk of radius \(\cosh ^{-1}(X/2)\) for \(\Gamma \) a discrete subgroup of \({\hbox {PSL}_2( {{\mathbb {R}}})} \). Selberg proved the estimate \(O(X^{2/3})\) 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 \({\hbox {PSL}_2( {{\mathbb {Z}}})} \). The result is that the error term can be improved to \(O(X^{7/12+{\varepsilon }})\). 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 \(O(X^{1/2+{\varepsilon }})\). In the appendix the relevant exponential sum over the spectral parameters is investigated.

Mathematics Subject Classification

Primary 11F72 Secondary 58J25 

Notes

Acknowledgments

The author of the appendix would like to thank Peter Sarnak for useful discussions and for providing notes for the co-compact case.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.Department of MathematicsUniversity College LondonLondonUK
  2. 2.Department of Mathematical SciencesUniversity of CopenhagenCopenhagenDenmark

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