Erdős-Kac type theorem for the number of scattering geodesics on modular surface

In 1917, Hardy and Ramanujan showed that if $\omega (n)$ is the number of distinct prime factors of a randomly chosen positive integer $n,$ then the normal order of $\omega (n)$ is $\log \log n\mathrm{.}$ This led Erdős and Kac to prove their celebrated result showing a Gaussian behaviour for $\omega (n)\mathrm{.}$ In this article we prove an Erdős-Kac kind result for the number of scattering geodesics on the modular surface with a common sojourn time.