<P> 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). </P> <P> 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 </P> <Dl> <Dd> E (2) = ∑ d = 1 ∞ d 6 π 2 d 2 = 6 π 2 ∑ d = 1 ∞ 1 d . (\ displaystyle \ mathrm (E) (\ mathrm (2)) = \ sum _ (d = 1) ^ (\ infty) d (\ frac (6) (\ pi ^ (2) d ^ (2))) = (\ frac (6) (\ pi ^ (2))) \ sum _ (d = 1) ^ (\ infty) (\ frac (1) (d)).) </Dd> </Dl> <Dd> E (2) = ∑ d = 1 ∞ d 6 π 2 d 2 = 6 π 2 ∑ d = 1 ∞ 1 d . (\ displaystyle \ mathrm (E) (\ mathrm (2)) = \ sum _ (d = 1) ^ (\ infty) d (\ frac (6) (\ pi ^ (2) d ^ (2))) = (\ frac (6) (\ pi ^ (2))) \ sum _ (d = 1) ^ (\ infty) (\ frac (1) (d)).) </Dd>

What is the greatest common divisor of m and n