# On Koyama's refinement of the prime geodesic theorem

## TL;DR

This paper presents a new proof for the prime geodesic theorem on compact Riemann surfaces, improving the error term without restrictive assumptions and extending the Gallagher-Koyama approach for better bounds.

## Contribution

It introduces a novel proof technique that refines the error term in the prime geodesic theorem without excluding any finite measure set, advancing previous results.

## Key findings

- Improved error term in the prime geodesic theorem.
- Elimination of the assumption excluding a finite logarithmic measure set.
- Extension of Gallagher-Koyama approach for stronger results.

## Abstract

We give a new proof of the best presently known error term in the prime geodesic theorem for compact Riemann surfaces, without the assumption of excluding a set of finite logarithmic measure. Stronger implications of the Gallagher-Koyama approach are derived yielding to a further reduction of the error term outside a set of finite logarithmic measure.