Spectral exponential sums on hyperbolic surfaces

Abstract

We study an exponential sum over Laplacian eigenvalues of Maaß forms on $$\varGamma \backslash {\mathbb {H}}$$ , where $$\varGamma $$ is a congruence subgroup of $${{\,\mathrm{SL}\,}}_{2}({\mathbb {Z}})$$ and $${\mathbb {H}}$$ is the upper half-plane. The goal is to establish an asymptotic formula that expresses the spectral exponential sum in terms of an oscillatory main term, the von Mangoldt function and the Selberg zeta function.