Niko Laaksonen (McGill)

Title:Prime Geodesic Theorem in the 3-dimensional Hyperbolic Space

Abstract: On hyperbolic manifolds the lengths of primitive closed geodesics (prime geodesics) have many similarities with the usual prime numbers. In particular, they obey an asymptotic distribution analogous to the Prime Number Theorem. The error in this estimation is well-studied in two dimensions. In three dimensions the only unconditional non-trivial estimate is by Sarnak. In this talk we show how to improve on Sarnak's pointwise bound for the error term. We also investigate the second moment of the error term and highlight some of the difficulties compared to the two dimensional case.