2020/10/31 by Squillace, Joseph
#11N37 #60F05 #FOS: Mathematics #Number Theory (math.NT) #Probability (math.PR)
paper · doi:10.48550/arxiv.2011.00152
Given n∈ℕ, let ω(n) denote the number of distinct prime factors of n, let Z denote a standard normal variable, and let Pn denote the uniform distribution on \ 1,…,n\ . The Erdős-Kac Theorem states that Pn(m≤ n:ω(m)-loglog n≤ x(loglog n)1/2)→ℙ(Z≤ x) as n→∞; i.e., if N(n) is a uniformly distributed variable on \lbrace 1,…,n \rbrace, then ω(N(n)) is asymptotically normally distributed as n→ ∞ with both mean and variance equal to log log n. The contribution of this paper is a generalization of the Erdős-Kac Theorem to a larger class of random variables by considering perturbations of the uniform probability mass (1)/(n) in the following sense. Denote by ℙn a probability distribution on \ 1,…,n\ given by ℙn(i)=(1)/(n)+εi,n. By providing some constraints on the εi,n's, sufficient conditions are stated in order to conclude that ℙn(m≤ n:ω(m)-loglog n≤ x(loglog n)1/2) → ℙ(Z≤ x) as n→∞. The main result will be applied to prove that the number of distinct prime factors of a positive integer with either the Harmonic(n) distribution or the Zipf(n,s) distribution also tends to the normal distribution N(loglog n,loglog n) as n→∞ (and as s→1 in the case of a Zipf variable).