On the Logarithmic Prime Geodesic Theorem
Abstract: In this paper we refine the error term in the prime geodesic theorem for compact, even-dimensional locally symmetric Riemannian manifolds with strictly negative sectional curvature. The ingredients for the starting prime geodesic theorem come from our most recent research of the zeta functions of Selberg and Ruelle associated with locally homogeneous bundles over compact locally symmetric spaces of rank one. In this paper, we shall restrict our investigations to compact, even-dimensional, locally symmetric spaces. For this class of spaces, we prove that there exists a set ∇ of positive real numbers, which is relatively small in the sense that its logarithmic measure is finite, such that the error term of the aforementioned prime geodesic theorem is improved outside ∇. The derived prime geodesic theorem generalizes the corresponding prime number theorem, where it is proved that the error term under Riemann hypothesis assumption can be further reduced except on a set of finite logarithmic measure.