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.
Categories: math.NT