Local average in hyperbolic lattice point counting

Abstract

The hyperbolic lattice point problem asks to estimate the size of the orbit $\Gamma z$ inside a hyperbolic disk of radius $\cosh^{-1}(X/2)$ for $\Gamma$ a discrete subgroup of $\hbox{PSL}_2(R)$. Selberg proved the estimate $O(X^{2/3})$ for the error term for cofinite or cocompact groups. This has not been improved for any group and any center. In this paper local averaging over the center is investigated for $\hbox{PSL}_2(Z)$. The result is that the error term can be improved to $O(X^{7/12+\epsilon})$. The proof uses surprisingly strong input e.g. results on the quantum ergodicity of Maa{\ss} cusp forms and estimates on spectral exponential sums. We also prove omega results for this averaging, consistent with the conjectural best error bound $O(X^{1/2+\epsilon})$. In the appendix the relevant exponential sum over the spectral parameters is investigated.