Multiplicative arithmetic functions and the Ewens measure
Dor Elboim, Ofir Gorodetsky · arXiv (Cornell University) · 2019
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 theorem on the largest prime factors of a random integer. 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 $\alpha(n)$. When the mean value of $\alpha$ over primes tends to some positive $\theta$, we draw an analogy between these measures and the Ewens measure with parameter $\theta$. In particular, we extend the results of Erdős-Kac and Billingsley to these measures, obtaining results which are in perfect agreement with the ones known for the Ewens measure. We also study a generalized Ewens measure on the integers arising from multiplicative $\alpha$ with $\alpha(p)=(\log p)^{\gamma}$, and obtain results in agreement with those in the permutation setting.