Olga Balkanova (Chalmers University of Technology and University of Gothenburg): Bounds for the spectral exponential sum

Abstract. Exponential sums over eigenvalues of the hyperbolic Laplacian for PSL(2,Z), called spectral exponential sums, are closely related to some classical problems in analytic number theory, including prime geodesic theorem and hyperbolic lattice point counting. Recently, Y. N. Petridis and M. S. Risager conjectured that the order of magnitude of such sums should be the square root of the number of terms. We show that this conjecture holds in some ranges by proving two new upper bounds for the spectral exponential sums. This is joint work with D. Frolenkov.