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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


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.


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

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