# On the prime geodesic theorem for hyperbolic 3-manifolds

## TL;DR

This paper improves the error term exponent in the prime geodesic theorem for hyperbolic 3-manifolds using the Selberg zeta approach, reducing it from 5/3 to 3/2 and further to 13/9 under certain conditions.

## Contribution

It introduces a new method to lower the error term exponent in the prime geodesic theorem for hyperbolic 3-manifolds, refining previous bounds.

## Key findings

- Exponent reduced from 5/3 to 3/2
- Further improved to 13/9 excluding a finite logarithmic measure set
- Enhanced understanding of prime geodesic distribution

## Abstract

Through the Selberg zeta approach, we reduce the exponent in the error term of the prime geodesic theorem for cocompact Kleinian groups or Bianchi groups from Sarnak's $\frac{5}{3}$ to $\frac{3}{2}$. At the cost of excluding a set of finite logarithmic measure, the bound is further improved to $\frac{13}{9}$.