Square root cancellation in the hyperbolic lattice counting problem over cocompact groups

For cocompact Fuchsian groups $\mathrm{\Gamma }\le \text{PSL}(2,\mathbb{R})$ we improve Selberg’s bound for the error term of the counting function in the hyperbolic lattice counting problem, achieving an essentially optimal upper bound for the error term \begin{equation*} E(X;z, w) = O(X^{1/2+\varepsilon}) \end{equation*} for almost every pair of points $z,w$. We extend this pointwise result for cocompact lattices acting on the $n$-dimensional real hyperbolic space ${\mathbb{H}}^n$. Moreover, in the $2$-dimensional case we study the second moment of the error term of the counting function. We improve the upper bounds of Chamizo and Cherubini for almost all pairs $z,w$, but we disprove a conjecture of Phillips and Rudnick for the case of cocompact Fuchsian groups.