Prime geodesic theorem of Gallagher type

TL;DR

This paper improves the error term in the prime geodesic theorem for compact Riemann surfaces, reducing the exponent from 3/4 to 7/10 outside a set of finite logarithmic measure.

Contribution

It provides a refined bound on the error term in the prime geodesic theorem, advancing the understanding of spectral geometry and number theory.

Findings

Error term exponent reduced from 3/4 to 7/10

Improved bounds hold outside a set of finite logarithmic measure

Advances in spectral analysis of Riemann surfaces

Abstract

We reduce the exponent in the error term of the prime geodesic theorem for compact Riemann surfaces from $\frac{3}{4}$ to $\frac{7}{10}$ outside a set of finite logarithmic measure.