THE PROBABILITY THAT RANDOM POSITIVE INTEGERS ARE k-WISE RELATIVELY PRIME

Abstract

The positive integers a1, a2, …, as are k-wise relatively prime if any k of them are relatively prime. For a (k-1)-tuple of positive integers u = (u1, …, uk-1), let [Formula: see text] denote the number of s-tuples of positive integers (a1, a2, …, as) with 1 ≤ a1, a2, …, as ≤ n such that a1, a2, …, as are k-wise relatively prime and are i-wise relatively prime to ui for i = 1, 2, …, k-1 when s ≥ k ≥ 2, and a1, a2, …, as are i-wise relatively prime to ui for i = 1, 2, …, s when k > s ≥ 1. Asymptotic estimates are obtained for these functions. As a corollary, exact formula is also obtained for the probability that s positive integers are k-wise relatively prime.