Multiplicative arithmetic functions and the generalized Ewens measure

Journal article

Random integers, sampled uniformly from [1, x], share similarities with random permutations, sampled uniformly from Sn. These similarities include the Erd˝os–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 nonuniform 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 {θi}i≥1 of non-negative reals, one may associate with it a measure on Sn 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.

Authors: Elboim, D; Gorodetsky, O

• Publication status: Published
• Peer review status: Peer reviewed
• Publisher: Springer
• Journal: Israel Journal of Mathematics
• Pages: 1-47
• Publication date: 2024-04-24
• Acceptance date: 2022-04-19
• DOI: 10.1007/s11856-024-2609-x
• EISSN: 1565-8511
• ISSN: 0021-2172
• Language: English