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].