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

TL;DR

This paper investigates the lower bounds of the error term in the hyperbolic lattice point problem on Riemann surfaces, extending previous work by analyzing the normalized error in a different parameter and establishing Omega results.

Contribution

It introduces new Omega results for the normalized error term in the hyperbolic lattice point problem, even when the center and orbit points differ.

Findings

Proves Omega-plus and Omega-minus bounds for the normalized error term.

Extends previous results to the case where the center and orbit points are different.

Analyzes the error term in the natural parameter X=2 cosh r.

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 $z \neq w$.