Dirichlet law for factorization of integers, polynomials and permutations

Sun-Kai Leung

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arXiv · 2206.14728

Dirichlet law for factorization of integers, polynomials and permutations

Sun-Kai Leung

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Abstract

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

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Sun-Kai Leung. 2022-06-29. Dirichlet law for factorization of integers, polynomials and permutations. https://arxiv.org/abs/2206.14728

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