# Mean square in the prime geodesic theorem

## TL;DR

This paper establishes improved average bounds for the remainder in the prime geodesic theorem across cofinite Fuchsian groups, utilizing trace formulas to enhance understanding of geodesic distribution.

## Contribution

It introduces new average bounds for the prime geodesic theorem's remainder, applying the Selberg and Kuznetsov trace formulas for broader classes of groups.

## Key findings

- Improved mean square bounds for the prime geodesic theorem's remainder.
- Enhanced bounds specifically for the modular group using Kuznetsov trace formula.
- Results surpass previous pointwise bounds on average.

## Abstract

We prove upper bounds for the mean square of the remainder in the prime geodesic theorem, for every cofinite Fuchsian group, which improve on average on the best known pointwise bounds. The proof relies on the Selberg trace formula. For the modular group we prove a refined upper bound by using the Kuznetsov trace formula.