PRIME GEODESIC THEOREM IN $mathbb{H}^3$ (Automorphic forms, automorphic representations and related topics)

Abstract

In these notes we give as ummary of the main developments in the Prime Geodesic Theorem on hyperbolic three-manifolds since the seminal result of Sarnak (1983). We will focus on arithmetic manifolds and show how strong tools, such as the Kuznetsov formula, allow us to improve on Sarnak's result in this case. In particular, it is currently known that the remainder in the Prime Geodesic Theorem on the Picard manifold is bounded by O(X3/2+Є) under Lindelöf hypothesis for quadratic Hecke L-functions over Gaussian integers. We will prove that this is true on average. The main results presented here appeared in a preprint of Chatzakos, Cherubini and Laaksonen (2018).