Author:
Elboim Dor,Gorodetsky Ofir
Abstract
AbstractRandom integers, sampled uniformly from [1, x], share similarities with random permutations, sampled uniformly from Sn. 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: 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.
Publisher
Springer Science and Business Media LLC
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