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

Abstract

Let {pj } ∞ j=1 denote the set of prime numbers in increasing order, let ΩN ⊂ N denote the set of positive integers with no prime factor larger than pN and let PN denote the probability measure on ΩN which gives to each n ∈ ΩN a probability proportional to 1 n . This measure is in fact the distribution of the random integer IN ∈ ΩN definedare independent random variables and Xp j is distributed as Geom(1 − 1 p j ). We show that log n log N under PN converges weakly to the Dickman distribution. As a corollary, we recover a classical result from multiplicative number theory-Mertens'We show that the two densities coincide on a natural algebra of subsets of N. We also show that they do not agree on the sets of n 1 s -smooth numbers {n ∈ N : p + (n) ≤ n 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 1 s -rough numbers {n ∈ N : p − (n) ≥ n 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, . . . , N } and under the PN -distribution on ΩN , 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.

On the strange domain of attraction to generalized Dickman distributions for sums of independent random variables

“…The examples in the above paragraph lead to limiting distributions where the Dickman function arises as a distribution function, not as a density as is the case with the GD(θ) distributions discussed in this paper. The GD(θ) distribution arises as a normalized limit in the context of certain natural probability measures that one can place on N; see [3], [7].…”

Probabilistic Proofs of Some Generalized Mertens' Formulas Via Generalized Dickman Distributions

“…One of the results in [5] involved the construction of a sequence of random integers whose distributions were shown to converge weakly to the so-called Dickman distribution. It was noted in that paper that Mertens' formula follows readily as a corollary of this result.…”

Dickman approximation in simulation, summations and perpetuities

“…We now present the proof of our first main result. Proof of Theorem 1.1: Let W n be as in (9) and take θ = 1 in (27). Letting…”

“…[27]) Π n by Π n (m) = 1 π n m for m ∈ Ω n with normalizing constant necessarily satisfying π n = m∈Ωn 1/m. Distributional convergence of S n to the standard Dickman distribution was proved in [27]. In Theorem 1.3 below, we provide (log n) −1 convergence rate in the Wasserstein-2 norm.…”