Erd\H{o}s-Kac type theorem for the number of scattering geodesics on modular surface

TL;DR

This paper proves an Erdős-Kac type theorem demonstrating Gaussian distribution behavior for the number of scattering geodesics with common sojourn time on the modular surface, extending classical number theory results to geometric dynamics.

Contribution

It establishes a new probabilistic distribution result for scattering geodesics, linking number theory and geometric dynamics in the context of the modular surface.

Findings

Number of scattering geodesics follows a Gaussian distribution.

Extension of Erdős-Kac theorem to geometric setting.

Provides probabilistic insights into geodesic behavior.

Abstract

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 $\log \log \, n.$ This led Erd\H{o}s and Kac to prove their celebrated result showing a Gaussian behaviour for $\omega(n).$ In this article we prove an Erd\H{o}s-Kac kind result for the number of scattering geodesics on the modular surface with a common sojourn time.