Does this relative primes formula violate the feasibility of picking truly random numbers?

Question

I read a question on this site recently that fascinated me by pointing out that you can't truly pick a random number from an infinite set. I can't find the answer now, but it was shown that you have to have a known distribution to achieve randomness. (Picking a number with all random digits was said to be infinitely more impossible because you could have an infinite number of digits before and after the decimal point.)

I'm not challenging that assertion; I believe it to be true.

However, I was recently reading about relative primes and came across this statement:

The probability that two integers m and n picked at random are relatively prime is 0.60792...

This raised several questions:

• What does this formula mean? I don't understand the math other than a vague conception of what the Riemann Zeta function is. Why is the function value of 2 significant to randomly chosen integers? And why take the inverse of it? The significance of 2 seems to stem from the number of integers being inspected, but why?

• How can you draw a conclusion about the relation of two randomly chosen integers when you can't truly choose a random integers from $-\infty$ to $+\infty$? Would it be more accurate to say the formula calculates the probability that two arbitrary integers will be relatively prime? I think this question might be better phrased as: How can you analyze all members of an infinite set?

Instant +1 to any answer with a visual depiction, as that seems to be the only way a math concept sticks in my brain.

Answer 1

You are right that the statement as it stands is imprecise. What it means is this:

Let $P_k$ be the probability that integers $m$ and $n$ are relatively prime, if $m$ and $n$ are each chosen at random from a uniform distribution on the set $\{1,2,\ldots,k\}$. Then

$$\lim_{k \to \infty} P_k = \frac{6}{\pi^2}$$

Alternatively, without using probability:

Let $N_k$ be the number of pairs $(m,n)$ in the set $\{1,2,\ldots,k\} \times \{1,2,\ldots,k\}$ that are relatively prime. Then

$$\lim_{k \to \infty} \frac{N_k}{k^2} = \frac{6}{\pi^2}$$