$\Omega$-results for the hyperbolic lattice point problem

Abstract

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

arXiv commentFinal version of arXiv:1512.04137 to appear in Proceedings of the AMS