Similar to the way I "computed" the probability that a "random pos. int."

is quadratfrei, the probability that two "random pos. ints." can also be computed.

 

Call the two pos. ints. K and L.

 

Now (K,L) = 1 <=> For each prime p, it is false that p divides both K and L.

 

But Prob(p divides a random integer N) = 1/p.

 

Assuming "independence" between the prime factors of K and those of L,

 

Prob(p divides each of K and L) = 1/p^2, and so

 

Prob(It is false that p divides each of K and L) = 1 - 1/p^2.

 

Thus, Prob(For each prime p, it is false that p divides each of K and L) =

 

= Product over all primes p of (1 - 1/p^2).

 

Once again this product is the reciprocal of the Euler product for Zeta(2),

and so the desired probability is 6/pi^2.

********************************************************************

 

NOW, how about three random integers K, L, M?

 

By same reasoning, for each p there must be at most one of the three random

integers that p divides. The probability of this is

 

Prob(p divides none of K,L,M) + Prob(p divides just one of K,L,M) =

 

(1-1/p)^3 + 3(1/p)*(1-1/p)^2 =

 

1 - 3/(p^2) + 2/(p^3).

 

Thus Prob(K,L,M) are relatively prime =

 

Product over all prime p of (1 - 3/(p^2) + 2/(p^3)).

 

I don't know how to sum this in closed form, but

numerically it turns out that

 

Prob( (K,L,M) = 1 ) ~ 0.1284

 

(taking the product over the first 1000 primes).

 

It would be nice to know that this jibes with the fraction of random triples

of pos. ints. that are rel. prime.