Niko Laaksonen: Prime Geodesic Theorem in the three-dimensional hyperbolic space

The Prime Geodesic Theorem (PGT) states that the lengths of primitive
closed geodesics on a hyperbolic manifold have an asymptotic behaviour
analogous to the usual prime numbers. Through an explicit formula for
the Selberg zeta function we can relate the error term to certain
spectral exponential sums. In the past few years there has been a
renewed interest in this problem especially in two and three
dimensions. In this talk we will outline some recent progress on the
pointwise and second moment bounds of the error term in the PGT on
various three dimensional hyperbolic manifolds.