For uniformly random q ≤ x, ω(n_q), the number of distinct prime factors of the count of scattering geodesics, is asymptotically normal with mean (1/2)(log log x)^2 and variance (1/3)(log log x)^3.
Kable, A variation of the Ikehara-Delange Tauberian theorem and an application, Comment
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Erd\H{o}s-Kac type theorem for the number of scattering geodesics on modular surface
For uniformly random q ≤ x, ω(n_q), the number of distinct prime factors of the count of scattering geodesics, is asymptotically normal with mean (1/2)(log log x)^2 and variance (1/3)(log log x)^3.