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

Theorems & Definitions (52)

• Theorem 1.1
• Theorem 1.2: Hansen, Watterson
• Theorem 1.3: Ercolani and Ueltschi
• Theorem 1.4: Ercolani and Ueltschi
• Definition 1.5
• Theorem 1.6
• Theorem 1.7
• Proposition 2.1
• Proposition 2.2
• proof