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Joint Poisson distribution of prime factors in sets
Abstarct Given disjoint subsets T 1 , …, T m 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 T 1 , …, T m 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 T 1 , …, T m 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.
Authors:
Award ID(s):
Publication Date:
NSF-PAR ID:
10338312
Journal Name:
Mathematical Proceedings of the Cambridge Philosophical Society
Volume:
173
Issue:
1
Page Range or eLocation-ID:
189 to 200
ISSN:
0305-0041
Sponsoring Org:
National Science Foundation
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