Greatest common divisor
Probabilities and expected value

In 1972, James E. Nymann showed that k integers, chosen independently and uniformly from {1,..., n }, are coprime with probability 1/ ζ (k) as n goes to infinity, where ζ refers to the Riemann zeta function. (See coprime for a derivation.) This result was extended in 1987 to show that the probability that k random integers have greatest common divisor d is d /ζ(k).

Using this information, the expected value of the greatest common divisor function can be seen (informally) to not exist when k = 2. In this case the probability that the GCD equals d is d / ζ (2), and since ζ (2) = π /6 we have

This last summation is the harmonic series, which diverges. However, when k ≥ 3, the expected value is well-defined, and by the above argument, it is

For k = 3, this is approximately equal to 1.3684. For k = 4, it is approximately 1.1106.
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