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Joint Poisson distribution of prime factors in sets

Part of: Multiplicative number theory

Published online by Cambridge University Press:  23 June 2021

KEVIN FORD
KEVIN FORD*
Affiliation:
Department of Mathematics, 1409 West Green Street, University of Illinois at Urbana-Champaign, Urbana, IL 61801, U.S.A. e-mail: ford@math.uiuc.edu
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Abstarct

Given disjoint subsets T1, …, Tm of “not too large” primes up to x, we establish that for a random integer n drawn from [1, x], the m-dimensional vector enumerating the number of prime factors of n from T1, …, Tm converges to a vector of m independent Poisson random variables. We give a specific rate of convergence using the Kubilius model of prime factors. We also show a universal upper bound of Poisson type when T1, …, Tm are unrestricted, and apply this to the distribution of the number of prime factors from a set T conditional on n having k total prime factors.

MSC classification

Primary: 11N25: Distribution of integers with specified multiplicative constraints
Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 173 , Issue 1 , July 2022 , pp. 189 - 200
Copyright
© The Author(s), 2021. Published by Cambridge University Press on behalf of Cambridge Philosophical Society

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Footnotes

†

Supported by NSF grant DMS-1802139.

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