# On Kuznetsov-Bykovskii's formula of counting prime geodesics

## TL;DR

This paper generalizes Kuznetsov-Bykovskii's formula for counting prime geodesics, extending its applicability to any number field and congruence subgroup, with explicit computations for specific cases.

## Contribution

It broadens the scope of the prime geodesic counting formula to general number fields and subgroups, providing explicit calculations for principal and Hecke subgroups.

## Key findings

- The generalized formula applies to any number field and congruence subgroup.
- Explicit computations are provided for principal and Hecke subgroups.
- The method enhances the understanding of prime geodesic distribution in various settings.

## Abstract

We generalize a formula on the counting of prime geodesics, due to Kuznetsov-Bykovskii, used in the work of Soundararajan-Young on the prime geodesic theorem. The method works over any number field and for any congruence subgroup. We give explicit computation in the cases of principal and Hecke subgroups.