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- W4288125310 abstract "Random integers, sampled uniformly from $[1,x]$, share similarities with random permutations, sampled uniformly from $S_n$. These similarities include the ErdH{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." @default.
- W4288125310 created "2022-07-28" @default.
- W4288125310 creator A5015315313 @default.
- W4288125310 creator A5056542452 @default.
- W4288125310 date "2019-09-02" @default.
- W4288125310 modified "2023-10-17" @default.
- W4288125310 title "Multiplicative arithmetic functions and the generalized Ewens measure" @default.
- W4288125310 doi "https://doi.org/10.48550/arxiv.1909.00601" @default.
- W4288125310 hasPublicationYear "2019" @default.
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