ON THE GAP DISTRIBUTION OF PRIME FACTORS

Let {pj (n)}(w(n)) j=1 denote the increasing sequence of distinct prime factors of an integer n. For z >= 0, let G(n; z) denote the number of those indexes j such that pj+1(n) > pj (n)(exp) (z). We show uniform convergence, with almost optimal effective estimate of the speed, of the distribution of G(n; z) on {n : 1 <= n <= N} to a Gaussian limit law with mean e(-z) log(2) (n), variance {e(-z) -2z e(-2z)} log(2) (n), and we establish an asymptotic formula with remainder for all centered moments.