Chebotarev geodesic theorem: split case

Abstract

We study the prime geodesic theorem in congruence classes of $\mathrm{SL}_2(\mathbb{Z})$. We generalize previous work of Luo and Sarnak and of Soundararajan and Young, and prove that the geodesic analogue of the Chebotarev density theorem holds with exponent $25/36 + \varepsilon$. In particular, we deduce that the prime geodesic theorem holds with exponent $25/36 + \varepsilon$ for any congruence subgroup of $\mathrm{SL}_2(\mathbb{Z})$.