Multiplicative arithmetic functions and the generalized Ewens measure
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arXiv
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| Format: | Preprint |
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2019
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| _version_ | 1866916420938366976 |
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| author | Elboim, Dor Gorodetsky, Ofir |
| author_facet | Elboim, Dor Gorodetsky, Ofir |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1909_00601 |
| institution | arXiv |
| publishDate | 2019 |
| record_format | arxiv |
| spellingShingle | Multiplicative arithmetic functions and the generalized Ewens measure Elboim, Dor Gorodetsky, Ofir Number Theory Probability 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. |
| title | Multiplicative arithmetic functions and the generalized Ewens measure |
| topic | Number Theory Probability |
| url | https://arxiv.org/abs/1909.00601 |