On the probability that two positive integers are relatively prime

In many of the sources I have consulted about this, the "probability" that two positive integers chosen at random are relatively prime is calculated as the limit as $n \to \infty$ of the probability that two randomly chosen integers in the set {1,2, ..., $n$} are relatively prime (the limit being $1/\zeta(2)$). My first question is: Is this limit really a probability?

Also, the nonrigorous/heuristic proofs that I have seen of this start by mentioning that "the probability that a prime $p$ divides a positive integer is $1/p$". This makes intuitive sense. I was wondering though: Is there a way of defining a probability measure on the positive integers in such a way that the set {$n \in \mathbf Z_+$ | $p$ divides $n$} has measure $1/p$ (that we can use for a rigorous proof)?

Solution 1:

The answer is yes, there is such a measure, if we weaken the notion of measure suitably. (The "if we weaken" part is a way of saying that this does not really answer the question.)

What is given up is countable additivity. Finite additivity is kept, and, very importantly, we can have translation invariance. (Obviously we cannot have both countable additivity and translation invariance. Since we can't have both, translation-invariance may be the more attractive property.)

There is a translation-invariant finitely additive "measure" on the positive integers that gives measure $1$ to $\mathbb{N}$, and that is defined on all subsets of $\mathbb{N}$. The idea goes back to Banach. One could look under Banach mean, invariant mean, or amenable (group or semigroup). There is a large literature.

The translation invariance guarantees that for any modulus $m$, and any integer $a$, the set of numbers congruent to $a$ modulo $m$ has "measure" $1/m$.

Construction of a Banach mean requires the Axiom of Choice.

Addendum: One can build stronger properties into a Banach mean. Let $A$ be any subset of $\mathbb{N}$, and let $A_n$ be the set of all $x \in A$ such that $x \le n$. If

$$\lim_{n\to\infty} \frac{|A_n|}{n}$$

exists, call that limit $d(A)$. There is a Banach mean that assigns "measure" $d(A)$ to any $A$ for which $d(A)$ exists.

Solution 2:

There are some probability measures on the set of integers, but quite often they do not correspond to our intuition (they are not translation invariant for example). To measure sets in number theory, we often rely on densities such as the natural density, of which you are speaking, and the analytic density, which is actually better behaved. The good news is when a set has both densities, they are equal.

Solution 3:

One may use the honest probability $P(X=n)=n^{-s}/\zeta(s)$ for $s>1$. From this honest probability one may prove that:

1 Events $E_{p}=\{X$ is divisible by $p\}$ are independent.

2 $P(gcd(X,Y)=n)=n^{-2s}/\zeta(2s)$ for independent $X$ and $Y$. If $s\to 1$ for $n=1$ we obtain the claim.

Some other results may be obtained. See ex. 4.2. in Probability with martingales by Williams.