Mean value results and -results for the hyperbolic lattice point problem in conjugacy classes

  • Dimitrios Chatzakos

    Université des Sciences et Technologies de Lille et CEMPI, Villeneuve-d'Ascq, France
Mean value results and $\Omega$-results for the hyperbolic lattice point problem in conjugacy classes cover
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Abstract

For a cofinite Fuchsian group, we study the lattice point problem in conjugacy classes on the Riemann surface . Let be a hyperbolic conjugacy class in and the -invariant closed geodesic on the surface. The main asymptotic for the counting function of the orbit inside a circle of radius centered at grows like . This problem is also related with counting distances of the orbit of from the geodesic . For we study mean value and -results for the error term of the counting function. We prove that a normalized version of the error has finite mean value in the parameter . Further, we prove that if is cocompact then \begin{align*} \int_{\ell} e(\mathcal{H}, X;z)\, d s(z) = \Omega \big( X^{1/2} \log \log \log X \big). \end{align*} For we prove the same -result, using a subconvexity bound for the Epstein zeta function associated to an indefinite quadratic form in two variables. We also study pointwise -results for the error term. Our results extend the work of Phillips and Rudnick for the classical lattice problem to the conjugacy class problem.

Cite this article

Dimitrios Chatzakos, Mean value results and -results for the hyperbolic lattice point problem in conjugacy classes. Rev. Mat. Iberoam. 35 (2019), no. 4, pp. 1123–1152

DOI 10.4171/RMI/1080