THE TUPLES OF NATURAL NUMBERS AND PRIME DIVISORS

Let P-1(n) >= P-2(n) >= ... be the prime divisors of a natural number n arranged in the nonincreasing order. We consider the limit distribution of the sequences (log P-i (n(1) ... n(s))/log(n(1) ... n(s)), i >= 1) for tuples of coprime natural numbers < n(1), ... ,n(s)> is an element of N((x) over right arrow). We prove that, under some conditions on the sets N((x) over right arrow), the limit distribution of the sequences exists and is closely related to the Poisson-Dirichlet distribution.