On the probability that k positive integers are relatively prime
Abstract
Let Pk(n) denote the probability that k positive integers, chosen at random from {1, 2,..., n}, are relatively prime. It is shown that Pk(n) = 1/[zeta](k) + O(1/n) if k >= 3 and P2(n) = 1/[zeta](2) + O(log n/n), where [zeta] denotes the Riemann [zeta]function. Hence for k >= 2, limn>[infinity] Pk(n) = 1/[zeta](k). The same problem is studied using probability distributions on the positive integers other than the uniform distribution on {1, 2,..., n} as was used above. The following result, with examples, is given: Let f be a probability density function, defined on the cartesian product of the set of positive integers with itself k times, which has the following property: if (m1,..., mk) = d, then f(m1,..., mk) = g(d)f(m1/d,..., mk/d) for some function g defined on the positive integers. Then the probability that a ktuple of positive integers chosen from this distribution is relatively prime is 1/[Sigma]d = 1[infinity] g(d).
 Publication:

Journal of Number Theory
 Pub Date:
 October 1972
 DOI:
 10.1016/0022314X(72)900388
 Bibcode:
 1972JNT.....4..469N