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 multiplicative number theory—Mertens’ formula. Let \(D_{\text {nat}}(A)\) denote the natural density of \(A\subset \mathbb {N}\), if it exists, and let \(D_{\text {log-indep}}(A)=\lim _{N\rightarrow \infty }P_N(A\cap \Omega _N)\) 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 \(p^+(n)\) denotes 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 \(p^-(n)\) denotes 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.