A Natural probabilistic model on the integers and its relation to Dickman-type distributions and Buchstab’s function

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 $\frac{1}{n}$. 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$(1-\frac{1}{p_j})$. We show that $\frac{\log n}{\log N}$ under $P_N$ converges weakly to the Dickman distribution. As a corollary, we recover a classical result from classical multiplicative number theory—Mertens’ formula, which states that $\sum_{n\in\Omega_N}\frac{1}{n}\sim e^\gamma\log N$ as $N\to\infty$.

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 $n^\frac{1}{s}$- smooth numbers $\{n\in\mathbb{N}: p^+(n)\le n^\frac{1}{s}\}$, $s>1$, 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 $n^\frac{1}{s}$- rough numbers ${n\in\mathbb{N}:p^-(n)\ge n^{\frac{1}{s}}}$, $s>1$, 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.