Multiplicative arithmetic functions and the generalized Ewens measure

Abstract

Random integers, sampled uniformly from [1, x], share similarities with random permutations, sampled uniformly from S_n. These similarities include the Erdős–Kac theorem on the distribution of the number of prime factors of a random integer, and Billingsley’s theorem on the largest prime factors of a random integer. In this paper we extend this analogy to non-uniform distributions.

Given a multiplicative function α: ℕ → ℝ_≥0, one may associate with it a measure on the integers in [1, x], where n is sampled with probability proportional to the value α(n). Analogously, given a sequence {θ_i}_i≥1 of non-negative reals, one may associate with it a measure on S_n that assigns to a permutation a probability proportional to a product of weights over the cycles of the permutation. This measure is known as the generalized Ewens measure.

We study the case where the mean value of α over primes tends to some positive θ, as well as the weights α(p) ≈ (log p)^γ. In both cases, we obtain results in the integer setting which are in agreement with those in the permutation setting.