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An all-purpose Erdös-Kac theorem

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Abstract

In a seminal paper of 1917, Hardy and Ramanujan showed that the normal number of prime factors of a random natural number n is \(\log \log n\). Their paper is often seen as inspiring the development of probabilistic number theory in that it led Erdös and Kac to discover, in 1940, a Gaussian law implied by their work. In this paper, we derive an all-purpose Erdös-Kac theorem that is applicable in diverse settings. In particular, we apply our theorem to show the validity of an Erdös-Kac type theorem for the study of the number of prime factors of sums of Fourier coefficients of Hecke eigenforms.

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Acknowledgements

We thank Siddhi Pathak and the referee for their helpful remarks on earlier versions of this paper

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Authors and Affiliations

  1. Department of Mathematics and Statistics, Queen’s University, Kingston, ON, K7L 3N6, Canada

    M. Ram Murty

  2. Department of Mathematics, University of Toronto, Toronto, ON, M5S 2E4, Canada

    V. Kumar Murty

  3. School of Mathematical Sciences, National Institute of Science Education and Research HBNI, Bhubaneswar P.O. Jatni, Khurda, Odisha, 752050, India

    Sudhir Pujahari

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  1. M. Ram Murty
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Correspondence to M. Ram Murty.

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Murty, M.R., Murty, V.K. & Pujahari, S. An all-purpose Erdös-Kac theorem. Math. Z. 305, 45 (2023). https://doi.org/10.1007/s00209-023-03370-y

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