Note on the prime divisors of Farey fractions

Abstract

Let P₁(n) ≥ P₂(n) ≥· · · be the prime divisors of a natural number n arranged in the non-increasing order. The limit distribution of the sequences (log P_i(mn)/ log(mn), i ≥ 1) for m/n ꞓ 2 (lambda₁; lambda₂), n ≤ x, are considered. It is proved that under some conditions on lambda_i the limit distribution of the sequences exists and is closely related to the Poisson–Dirichlet distribution.