A Natural Probabilistic Model on the Integers and its Relation to Dickman-Type Distributions and Buchstab's Function

TL;DR

This paper introduces a probabilistic model for integers based on prime factorization, demonstrating convergence to the Dickman distribution, comparing natural and probabilistic densities, and analyzing smooth and rough numbers with connections to the Buchstab function.

Contribution

It establishes a new probabilistic framework linking prime factorization distributions to Dickman and Buchstab functions, with novel convergence and density results.

Findings

Logarithm of integers under the model converges to the Dickman distribution.

Natural and probabilistic densities coincide on a natural algebra of sets.

Asymptotic decay profiles of smooth and rough numbers involve the Buchstab function.

Abstract

Let $\{p_j\}_{j=1}^\infty$ denote the set of prime numbers in increasing order, let $\Omega_N\subset \mathbb{N}$ denote the set of positive integers with no prime factor larger than $p_N$ and let $P_N$ denote the probability measure on $\Omega_N$ which gives to each $n\in\Omega_N$ a probability proportional to $\frac1n$. This measure is in fact the distribution of the random integer $I_N\in\Omega_N$ defined by $I_N=\prod_{j=1}^Np_j^{X_{p_j}}$, where $\{X_{p_j}\}_{j=1}^\infty$ are independent random variables and $X_{p_j}$ is distributed as Geom$(1-\frac1{p_j})$. We show that $\frac{\log n}{\log N}$ under $P_N$ converges weakly to the Dickman distribution. Let $D_{\text{nat}}(A)$ denote the natural density of $A\subset\mathbb{N}$, if it exists, and let $D_{\text{log-indep}}(A)=\lim_{N\to\infty}P_N(A\cap\Omega_N)$ denote the density of $A$ arising from $\{P_N\}_{N=1}^\infty$, if it exists. We show that the two densities coincide on a natural algebra of subsets of $\mathbb{N}$. We also show that they do not agree on the sets of $n^\frac1s$-\it smooth numbers \rm\ $\{n\in\mathbb{N}: p^+(n)\le n^\frac1s\}$, $s>1$, where $p^+(n)$ is the largest prime divisor of $n$. This last consideration concerns distributions involving the Dickman function. We also consider the sets of $n^\frac1s$-\it rough numbers \rm\ $\{n\in\mathbb{N}:p^-(n)\ge n^{\frac1s}\}$, $s>1$, where $p^-(n)$ is the smallest prime divisor of $n$. We show that the probabilities of these sets, under the uniform distribution on $[N]=\{1,\ldots, N\}$ and under the $P_N$-distribution on $\Omega_N$, have the same asymptotic decay profile as functions of $s$, although their rates are necessarily different. This profile involves the Buchstab function. We also prove a new representation for the Buchstab function.