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

Volume 192 / 2020

Daniel El-Baz 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.

Authors

  • Daniel El-BazSchool of Mathematical Sciences
    Tel Aviv University
    Tel Aviv, Israel
    and
    Max Planck Institute for Mathematics
    Vivatsgasse 7
    53111 Bonn, Germany
    e-mail

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