Spectral exponential sums on hyperbolic surfaces

We study an exponential sum over Laplacian eigenvalues of Maaß forms on Γ∖ℍ, where Γ is a congruence subgroup of SL₂(ℤ) and 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.