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Suppose you pick two random numbers less than n, then [n/2]2 pairs are both divisible by 2. [n/3]2 pairs are both divisible by 3. [n/5]2 pairs are both divisible by 5. ... (Here [x] is the greatest integer less than or equal to x, usually called the floor function.) So the number of relatively prime pairs less than or equal to n is (by the inclusion/exclusion principle): n 2 - sum([n/p]2) + sum([n/pq]2) - sum([n/pqr]2) + ... here the sums are taken over the primes p,q,r,... less than n. Letting mu(x) be the möbius function this is so the desired constant is the limit as n goes to infinity of this sum divided by n2, or sum(mu(k)/k2) (sum over positive integers k). But this series times the sum of the reciprocals of the squares is one, so the sum of this series, the desired limit, is 6/2. This number is approximately sum(mu(k)[n/k]2) (sum over positive integers k) 60.7927...%