TY - JOUR

T1 - Joint Poisson distribution of prime factors in sets

AU - Ford, Kevin

N1 - Publisher Copyright:
©

PY - 2022/7/23

Y1 - 2022/7/23

N2 - 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.

AB - 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.

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U2 - 10.1017/S0305004121000499

DO - 10.1017/S0305004121000499

M3 - Article

AN - SCOPUS:85108909373

SN - 0305-0041

VL - 173

SP - 189

EP - 200

JO - Mathematical Proceedings of the Cambridge Philosophical Society

JF - Mathematical Proceedings of the Cambridge Philosophical Society

IS - 1

ER -