$Ω$-results for the hyperbolic lattice point problem

For $Γ$ a cocompact or cofinite Fuchsian group, we study the lattice point problem on the Riemann surface $Γ\backslash\mathbb{H}$. The main asymptotic for the counting of the orbit $Γz$ inside a circle of radius $r$ centered at $z$ grows like $c e^r$. Phillips and Rudnick studied $Ω$-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 $Ω_{\pm}$-results for the orbit $Γw$ and circle centered at $z$, even for $z \neq w$.

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