On the asymptotic distribution of large prime factors

A random integer N, drawn uniformly from the set {1, 2,…, n}, has a prime factorization of the form N = α 1 α 2 …α M where α 1 ≥ α 2 ≥ … ≥ α M . We establish the asymptotic distribution, as n → ∞, of the vector A(n) = (log α 1 ,/log N: i: ≥ 1) in a transparent manner. By randomly re-ordering the components of A(n), in a size-biased manner, we obtain a new vector B(n) whose asymptotic distribution is the GEM distribution with parameter 1; this is a distribution on the infinite-dimensional simplex of vectors (x 1 , x 2 ,…) having non-negative components with unit sum. Using a standard continuity argument, this entails the weak convergence of A(n) to the corresponding Poisson–Dirichlet distribution on this simplex; this result was obtained by Billingsley [3]. © 1933, Oxford University Press. All rights reserved.