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

Abstract

For $\Gamma$ a Fuchsian group of finite covolume, we study the lattice point problem in conjugacy classes on the Riemann surface $\Gamma \backslash \mathbb{H}$. Let $\mathcal{H}$ be a hyperbolic conjugacy class in $\Gamma$ and $\ell$ the $\mathcal{H}$-invariant closed geodesic on the surface. The main asymptotic for the counting function of the orbit $\mathcal{H} \cdot z$ inside a circle of radius $t$ centered at $z$ grows like $c_{\mathcal{H}} \cdot e^{t/2}$. This problem is also related with counting distances of the orbit of $z$ from the geodesic $\ell$. For $X \sim e^{t/2}$ we study mean value and $\Omega$-results for the error term $e(\mathcal{H}, X ;z)$ of the counting function. We prove that a normalized version of the error $e(\mathcal{H}, X ;z)$ has finite mean value in the parameter $t$. Further, we prove that if $\Gamma$ is cocompact then \begin{eqnarray*} \int_{\ell} e(\mathcal{H}, X;z) d s(z) = \Omega \left( X^{1/2} \log \log \log X \right). \end{eqnarray*} We prove that the same $\Omega$-result holds for $\Gamma = {\hbox{PSL}_2( {\mathbb Z})}$ if we assume a subconvexity bound for the Epstein zeta function associated to an indefinite quadratic form in four variables. We also study pointwise $\Omega_{\pm}$-results for the error term. Our results extend the work of Phillips and Rudnick for the classical lattice problem to the conjugacy class problem.