Spectral Exponential Sums on Hyperbolic Surfaces

We study an exponential sum over Laplacian eigenvalues \(\lambda_{j} = 1/4+t_{j}^{2}\) with \(t_{j} \leqslant T\) for Maass cusp forms on \(\Gamma \backslash \mathbb{H}\), where \(\Gamma\) is a cofinite Fuchsian group acting on the upper half-plane \(\mathbb{H}\). The aim is to establish an asymptotic formula which expresses spectral exponential sums in terms of an oscillatory component, von Mangoldt-like functions and Selberg zeta functions. The behaviour is determined by whether \(\Gamma\) is essentially cuspidal or not.