What is the probability that gcd(n,m)=1? 
(from the Prime Pages' list of frequently asked questions)
 New record prime: 277,232,917-1 with 23,249,425 digits by Pace, Woltman, Kurowski, Blosser & GIMPS (26 Dec 2017).

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What is the probability that gcd(n,m)=1? Here is a frequently asked question
While playing with programs to determine primes and relative primes, I stumbled over an interesting (at least to me) fact. While the probability of a random number being prime decreases as the range of possible random numbers increases (Prime Number Theorem), the probability of two random numbers being relatively prime is 60.8% Is this something that is either well known by or trivially obvious to prime number gurus? 
That there should be such a constant is "obvious", finding its value takes more work. In 1849 Dirichlet showed that the probability is 6/pi2 roughly as follows. 

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): 
n2 - sum([n/p]2) + sum([n/pq]2) - sum([n/pqr]2) + ... 
where the sums are taken over the primes p,q,r,... less than n. Letting mu(x) be the möbius function this is 
sum(mu(k)[n/k]2)     (sum over positive integers k
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/pi2. This number is approximately 
percent.  ;-)
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