The Prime Geodesic Theorem

TL;DR

This paper improves the error term in the prime geodesic theorem for the modular surface and explores the distribution of prime geodesics, especially in short intervals, under the assumption of the generalized Riemann Hypothesis.

Contribution

It advances the understanding of prime geodesic distribution by refining error bounds and establishing conditional results linked to Dirichlet L-functions.

Findings

Improved error term in the prime geodesic theorem.

Conditional results on prime geodesics in very short intervals.

Established a connection between closed geodesics and Dirichlet L-functions.

Abstract

We study the distribution of closed geodesics for the modular surface. We improve the error term in the prime geodesic theorem, and obtain results on prime geodesics in very short intervals conditionally on the generalized Riemann Hypothesis for Dirichlet L-functions. We emphasize a connection between the closed geodesics and certain Dirichlet L-functions.