Prime scattering geodesic theorem

The modular surface, given by the quotient $\mathcal{M} = \Ha/\text{PSL}(2,\Z)$, can be partitioned into a compact subset $\Mm$ and an open neighborhood of the unique cusp in $\mathcal{M}$. We consider scattering geodesics in $\mathcal{M}$, first introduced by Victor Guillemin in \cite{Guillemin1976-xr} for hyperbolic surfaces with cusps. These are geodesics in $\mathcal{M}$ that lie in $\mathcal{M} \setminus \Mm$ for both large positive and negative times. Associated with such a scattering geodesic in $\mathcal{M}$, a finite \textit{sojourn time} is defined in \cite{Guillemin1976-xr}. In this article, we study the distribution of these scattering geodesics in $\mathcal{M}$ and their associated \textit{sojourn times}. In this process, we establish a connection between the counting of scattering geodesics on the modular surface and the study of positive integers whose prime divisors lie in arithmetic progression. This article is the first such result for scattering geodesics.