Ω-results for the hyperbolic lattice point problem

Chatzakos, Dimitrios

doi:10.1090/proc/13348

$\Omega$-results for the hyperbolic lattice point problem
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by Dimitrios Chatzakos PDF

Proc. Amer. Math. Soc. 145 (2017), 1421-1437 Request permission

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 $z \neq w$.

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