Spectral Exponential Sums on Hyperbolic Surfaces

We study an exponential sum over Laplacian eigenvalues $\lambda_{j} = 1/4+t_{j}^{2}$with$t_{j} \leqslant T$forMaasscuspformson$\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 $\mathrm{\Gamma }$ is essentially cuspidal or not.