Consider the case for n=2. Now if we say two numbers are relatively prime , we mean they dont share a common prime factor(it is enough to check prime factors). In other words:(m,n)=1, the gcd of m and n =1 => p a prime doesn't divide m and p doesn't divide n. So p^2 doesn't divide mn. But as m and n range through positive integers they represent all the positive integers. So we can simply write mn=l, l belongs to Z+. But this is exactly the probability that a randomly chosen number is square free, which again is zeta(2). We can extend this to n=k.

Consider the case for k=2. Now if we say two numbers are relatively prime , we mean they dont share a common prime factor(it is enough to check prime factors). In other words:(m,n)=1, the gcd of m and n =1 => p a prime doesn't divide m and p doesn't divide n. So p^2 doesn't divide mn. But as m and n range through positive integers they represent all the positive integers. So we can simply write mn=l, l belongs to Z+. But this is exactly the probability that a randomly chosen number is square free, which again is zeta(2). We can extend this to k>2.