$\mathrm{\Omega }$-results for the hyperbolic lattice point problem

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