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$ \Omega$-results for the hyperbolic lattice point problem

Author: Dimitrios Chatzakos
Journal: Proc. Amer. Math. Soc. 145 (2017), 1421-1437
MSC (2010): Primary 11F72; Secondary 37C35, 37D40
Published electronically: October 24, 2016
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Abstract: For $ \Gamma $ a cocompact or cofinite Fuchsian group, we study the lattice point problem on the Riemann surface $ \Gamma \backslash \mathbb{H}$. The main asymptotic for the counting of the orbit $ \Gamma z$ inside a circle of radius $ r$ centered at $ z$ grows like $ c e^r$. Phillips and Rudnick studied $ \Omega $-results for the error term and mean results in $ r$ for the normalized error term. We investigate the normalized error term in the natural parameter $ X=2 \cosh r$ and prove $ \Omega _{\pm }$-results for the orbit $ \Gamma w$ and circle centered at $ z$, even for $ z \neq w$.

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  • [1] Bruce C. Berndt, Sun Kim, and Alexandru Zaharescu, Circle and divisor problems, and double series of Bessel functions, Adv. Math. 236 (2013), 24-59. MR 3019715,
  • [2] Fernando Chamizo, Some applications of large sieve in Riemann surfaces, Acta Arith. 77 (1996), no. 4, 315-337. MR 1414513
  • [3] Harald Cramér, Ein Mittelwertsatz in der Primzahltheorie, Math. Z. 12 (1922), no. 1, 147-153 (German). MR 1544509,
  • [4] Jean Delsarte, Sur le gitter fuchsien, C. R. Acad. Sci. Paris 214 (1942), 147-179 (French). MR 0007769
  • [5] I. S. Gradshteyn and I. M. Ryzhik, Table of integrals, series, and products, 7th ed., Elsevier/Academic Press, Amsterdam, 2007. Translated from the Russian; Translation edited and with a preface by Alan Jeffrey and Daniel Zwillinger; With one CD-ROM (Windows, Macintosh and UNIX). MR 2360010
  • [6] Anton Good, Local analysis of Selberg's trace formula, Lecture Notes in Mathematics, vol. 1040, Springer-Verlag, Berlin, 1983. MR 727476
  • [7] Paul Günther, Gitterpunktprobleme in symmetrischen Riemannschen Räumen vom Rang $ 1$, Math. Nachr. 94 (1980), 5-27 (German). MR 582516,
  • [8] G. H. Hardy, On the expression of a number as the sum of two squares, Quart. J. Math. 46, 263-283, 1915.
  • [9] G. H. Hardy, On Dirichlet's Divisor Problem, Proc. London Math. Soc. S2-15, no. 1, 1-g. MR 1576550,
  • [10] G. H. Hardy and E. M. Wright, An introduction to the theory of numbers, 6th ed., Oxford University Press, Oxford, 2008. Revised by D. R. Heath-Brown and J. H. Silverman; With a foreword by Andrew Wiles. MR 2445243
  • [11] B. Huang and Z. Xu. The sup-norm bounds for Eisenstein series. arXiv:1508.02799
  • [12] Heinz Huber, Zur analytischen Theorie hyperbolischen Raumformen und Bewegungsgruppen, Math. Ann. 138 (1959), 1-26 (German). MR 0109212
  • [13] M. N. Huxley, Scattering matrices for congruence subgroups, Modular forms (Durham, 1983) Ellis Horwood Ser. Math. Appl.: Statist. Oper. Res., Horwood, Chichester, 1984, pp. 141-156. MR 803366
  • [14] H. Iwaniec. Spectral methods of automorphic forms. Second edition. Graduate Studies in Mathematics, 53. American Mathematical Society, Providence, RI; Revista MatemÁtica Iberoamericana, Madrid, 2002. xii+220 pp.
  • [15] S. J. Patterson, A lattice-point problem in hyperbolic space, Mathematika 22 (1975), no. 1, 81-88. MR 0422160
  • [16] S. J. Patterson, Corrigendum to: ``A lattice-point problem in hyperbolic space'' (Mathematika 22 (1975), no. 1, 81-88), Mathematika 23, no. 2, 227, 1976.
  • [17] Ioannis Nicolaos Petridis, Scattering theory for automorphic functions and its relation to L-functions, ProQuest LLC, Ann Arbor, MI, 1993. Thesis (Ph.D.)-Stanford University. MR 2688663
  • [18] Ralph Phillips and Zeév Rudnick, The circle problem in the hyperbolic plane, J. Funct. Anal. 121 (1994), no. 1, 78-116. MR 1270589,
  • [19] A. Selberg, Equidistribution in discrete groups and the spectral theory of automorphic forms,
  • [20] M. Young. A note on the sup norm of Eisenstein series. arXiv:1504.03272

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Additional Information

Dimitrios Chatzakos
Affiliation: Department of Mathematics, University College London, Gower Street, London WC1E 6BT, United Kingdom

Received by editor(s): December 14, 2015
Received by editor(s) in revised form: May 11, 2016, and May 24, 2016
Published electronically: October 24, 2016
Additional Notes: The author was supported by a DTA from EPSRC during his Ph.D. studies at UCL
Communicated by: Kathrin Bringmann
Article copyright: © Copyright 2016 American Mathematical Society

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