  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.
