Dirichlet law for factorization of integers, polynomials and permutations

TL;DR

This paper demonstrates that the factorization of integers, polynomials, and permutations into k parts follows a Dirichlet distribution, extending known results to a broader class of mathematical objects through multidimensional contour integration.

Contribution

It generalizes the Deshouillers-Dress-Tenenbaum arcsine law to k-part factorizations and extends the Dirichlet distribution modeling to polynomials and permutations.

Findings

Factorization of integers into k parts follows Dirichlet distribution.

The result extends to polynomials and permutations.

Multidimensional contour integration is used to prove the distribution law.

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.