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Dirichlet law for factorisation of integers, polynomials and permutations

Part of: Multiplicative number theory Combinatorics Zeta and $L$-functions: analytic theory Finite fields and commutative rings (number-theoretic aspects)

Published online by Cambridge University Press:  06 September 2023

SUN–KAI LEUNG
SUN–KAI LEUNG*
Affiliation:
Département de Mathématiques et de Statistique, Université de Montréal, CP 6128 succ. Centre-Ville, Montréal, QC H3C 3J7, Canada. e-mail: sunkaileung@gmail.com
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Abstract

Let $k \geqslant 2$ be an integer. We prove that factorisation of integers into k parts follows the Dirichlet distribution $\mathrm{Dir}\left({1}/{k},\ldots,{1}/{k}\right)$ by multidimensional contour integration, thereby generalising the Deshouillers–Dress–Tenenbaum (DDT) arcsine law on divisors where $k=2$. The same holds for factorisation of polynomials or permutations. Dirichlet distribution with arbitrary parameters can be modelled similarly.

MSC classification

Primary: 11M32: Multiple Dirichlet series and zeta functions and multizeta values 11N37: Asymptotic results on arithmetic functions 11N60: Distribution functions associated with additive and positive multiplicative functions
Secondary: 05A05: Permutations, words, matrices 05A16: Asymptotic enumeration 11T06: Polynomials
Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 175 , Issue 3 , November 2023 , pp. 649 - 676
Copyright
© The Author(s), 2023. Published by Cambridge University Press on behalf of Cambridge Philosophical Society

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