Asymptotic behaviour of the spectral function of the Laplace–Beltrami operator for cocompact discrete subgroups of $\operatorname {SL}_2(\mathbb R)$

An asymptotic formula, uniform in $z$ and $z'$, is obtained for the spectral function $\theta (z,z',\lambda )$ of the Laplace–Beltrami operator for cocompact discrete subgroups of $\operatorname {SL}_2(\mathbb R)$ with power-law lowering of the order of the remainder.