# Prime geodesic theorem for the modular surface

## TL;DR

This paper improves the error term in the prime geodesic theorem for the modular surface under the generalized Lindelöf hypothesis, reducing the exponent to 5/8+ε outside a finite logarithmic measure set.

## Contribution

It provides a refined error estimate for the prime geodesic theorem assuming the generalized Lindelöf hypothesis, enhancing previous bounds.

## Key findings

- Error term exponent reduced to 5/8+ε
- Improvement holds outside a finite logarithmic measure set
- Conditional on the generalized Lindelöf hypothesis

## Abstract

Under the generalized Lindel\"{o}f hypothesis, the exponent in the error term of the prime geodesic theorem for the modular surface is reduced to $\frac{5}{8}+\varepsilon $ outside a set of finite logarithmic measure.