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 Erd\H{o}s--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 $\alpha \colon \mathbb{N} \to \mathbb{R}_{\ge 0}$, one may associate with it a measure on the integers in $[1,x]$, where $n$ is sampled with probability proportional to the value $\alpha(n)$. Analogously, given a sequence $\{ \theta_i\}_{i \ge 1}$ 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 $\alpha$ over primes tends to some positive $\theta$, as well as the weights $\alpha(p) \approx (\log p)^{\gamma}$. In both cases, we obtain results in the integer setting which are in agreement with those in the permutation setting.