An analogue of the Erdős-Kac theorem for the special linear group over the integers

Abstract

We investigate the number of prime factors of individual entries for matrices in the special linear group over the integers. We show that, when properly normalised, it satisfies a central limit theorem of Erdős-Kac-type. To do so, we employ a sieve-theoretic set-up due to Granville and Soundararajan. We also make use of an estimate coming from homogeneous dynamics due to Gorodnik and Nevo.

6 pages, incorporates the referee's corrections and suggestions; to appear in Acta Arithmetica