An analogue of the Erdős–Kac theorem for the special linear group over the integers
Volume 192 / 2020
Acta Arithmetica 192 (2020), 181-188
MSC: Primary 11N36; Secondary 22F30.
DOI: 10.4064/aa181121-26-3
Published online: 8 November 2019
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.
