On the probability that k positive integers are relatively prime

Abstract Given a set S of positive integers let Z k S (t) denote the number of k-tuples 〈m1, …, mk〉 for which m i ∈ S ▶ [1, t] and (m1, …, mk) = 1. Also let P k S (n) denote the probability that k integers, chosen at random from S ▶ [1, n] , are relatively prime. It is shown that if P = {p1, …, pr} is a finite set of primes and S = {m : (m, p1 … pr) = 1}, then Z k S (t) = (td(S)) k Π ν∉P (1 − 1 p k ) + O(t k−1 ) if k ≥ 3 and Z 2 S (t) = (td(S)) 2 Π p∉P (1 − 1 p 2 ) + O(t log t) where d(S) denotes the natural density of S. From this result it follows immediately that P k S (n) → Π p∉P (1 − 1 p k ) = (ζ(k)) −1 Π p∈P (1 − 1 p k ) −1 as n → ∞. This result generalizes an earlier result of the author's where P = ⊘ and S is then the whole set of positive integers. It is also shown that if S = {p1x1 … prxr : xi = 0, 1, 2,…}, then P k S (n) → 0 as n → ∞.