The prime geodesic theorem in square mean

TL;DR

This paper improves the understanding of the average error in the prime geodesic theorem for PSL(2,Z), providing stronger bounds and a version applicable to short intervals.

Contribution

It strengthens recent results on the square mean of the error term and develops a short interval version of the prime geodesic theorem.

Findings

Enhanced bounds on the error term in the prime geodesic theorem

Development of a short interval version of the theorem

Improved understanding of error distribution in geodesic counting

Abstract

We strengthen the recent result of Cherubini and Guerreiro on the square mean of the error term in the prime geodesic theorem for $\mathrm{PSL}_2(\mathbb{Z})$. We also develop a short interval version of this result.