The Prime Geodesic Theorem in Arithmetic Progressions

TL;DR

This paper investigates the distribution of prime geodesics on the modular surface within arithmetic progressions, resolving longstanding conjectures and demonstrating non-equidistribution in residue classes.

Contribution

It proves that traces of closed geodesics do not evenly distribute across residue classes, addressing conjectures from Golovchanski-Smotrv (1999).

Findings

Traces of closed geodesics are not equidistributed in residue classes.

Resolved conjectures related to prime geodesic distributions.

Provides new insights into the distribution properties of geodesics.

Abstract

We address the prime geodesic theorem in arithmetic progressions, and resolve conjectures of Golovchanski\u{\i}-Smotrov (1999). In particular, we prove that the traces of closed geodesics on the modular surface do not equidistribute in the reduced residue classes of a given modulus.