This work addresses the prime geodesic theorem for the Picard manifold , which asks for the asymptotic evaluation of a counting function for the closed geodesics on . Let be the error term in the prime geodesic theorem. We establish that on average as well as many pointwise bounds. The second moment bound parallels an analogous result for due to Balog et al. and our innovation features the delicate analysis of sums of Kloosterman sums with an explicit manipulation of oscillatory integrals. The proof of the pointwise bounds requires Weyl-strength subconvexity for quadratic Dirichlet -functions over . Moreover, an asymptotic formula for a spectral exponential sum in the spectral aspect for a cofinite Kleinian group is given. Our numerical experiments visualise in particular that obeys a conjectural bound of size .
Ikuya Kaneko. "The prime geodesic theorem for and spectral exponential sums." Algebra Number Theory 16 (8) 1845 - 1887, 2022. https://doi.org/10.2140/ant.2022.16.1845