For each real number $t>1$, one can define a probability distribution $P_{t} = \left\{\mathrm{pr}_{t}\left(n\right)\right\}_{n\in\mathbb{N}}$ by $\mathrm{pr}_{t}\left(n\right) = \displaystyle{\frac{n^{-t}}{\zeta\left(t\right)}}$. This class of probability functions was studied by Golomb in [1]. In this post, we will prove a theorem about relatively $r$-prime $s$-tuples of integers.
This problem was studied for the case $r = 1$ by Nymann in Example 2 of Section 2 of [2].
In the following, unless otherwise specified, we take $t$ to be any real number greater than $1,$ and any mention of random integers will be taken to be chosen according to the distribution $P_{t}$. For notational convenience, we will take $0^{-t}$ to be $0$, so $\mathrm{pr}_{t}\left(0\right) = 0$.
Lemma. Fix an integer $a\geq 1$. The probability that a random integer is divisible by $a$ is $a^{-t}$.
Proof. $$\mathrm{pr}_{t}\left(a\mathbb{N}\right) = \sum_{m\in a\mathbb{N}} \frac{m^{-t}}{\zeta\left(t\right)} = \sum_{n\geq 1} \frac{\left(an\right)^{-t}}{\zeta\left(t\right)} = \frac{a^{-t}}{\zeta\left(t\right)} \sum_{n\geq 1}n^{-t} = a^{-t}.$$ QED
Corollary. Fix integers $a,b\geq 1$. The probability that a random integer is divisible by both $a$ and $b$ is $\mathrm{lcm}\left(a,b\right)^{-t}$.
In particular, if $a$ and $b$ are relatively prime, then divisibility by $a$ and $b$ are independent events.
Proof.
Note that $a\mathbb{N}\cap b\mathbb{N} = \mathrm{lcm}\left(a,b\right)\mathbb{N}$.
When $a$ and $b$ are relatively prime, we have that $\mathrm{lcm}\left(a,b\right) = ab$, and, hence, the event in question has probability $\left(ab\right)^{-t} = a^{-t}b^{-t}$.
QED
In [1], Golomb used this fact to prove that $\zeta\left(t\right) = \prod\limits_{p}\left(1-p^{-t}\right)^{-1}$ by considering $\mathrm{pr}_{t}\left(1\right)$. Here, instead, we will look at my favorite problem, namely proving results about relative primality or freeness of integers chosen according to this class of distributions.
Proposition. Fix an integer $r\geq 1$. The probability that a random integer is $r$-free is $\displaystyle{\frac{1}{\zeta\left(rt\right)}}$.
Proof.
Let $A\subseteq\mathbb{N}$ be the set of $r$-free positive integers. An integer $n$ is $r$-free if and only if $p^{r}\nmid n$ for all primes $p$. Therefore, $$A = \bigcap_{p} \left\{n\in\mathbb{N} : p^{r}\nmid n\right\} = $$
$$\bigcap_{p}\left(\mathbb{N}\setminus p^{r}\mathbb{N}\right).$$
Hence, the probability of this event is $$\mathrm{pr}_{t}\left(A\right) = \prod_{p}\left(1 – p^{-rt}\right) = \frac{1}{\zeta\left(rt\right)},$$ as required.
QED
That was warm up for the following theorem.
Theorem. Fix integers $r,s\geq 1$. The probability that $s$ random integers are relatively $r$-prime is $1/\zeta\left(rst\right)$.
Proof.
Let $A\subseteq\mathbb{N}^{s}$ denote the set of $s$-tuples of integers $\left(a_{1},\ldots,a_{s}\right)$ such that $\gcd\left(a_{1},\ldots,a_{s}\right)$ is $r$-free. Then, $$A = \bigcap_{p}\left\{\left(a_{1},\ldots,a_{s}\right)\in\mathbb{N}^{s} : p^{r}\nmid \gcd\left(a_{1},\ldots,a_{s}\right)\right\} =$$
$$\bigcap_{p} \mathbb{N}^{s}\setminus \left\{\left(a_{1},\ldots,a_{s}\right)\in\mathbb{N}^{s} : p^{r}\mid \gcd\left(a_{1},\ldots,a_{s}\right)\right\} = $$
$$\bigcap_{p} \mathbb{N}^{s} \setminus \left\{\left(a_{1},\ldots,a_{s}\right)\in\mathbb{N}^{s} : p^{r}\mid a_{i}\text{ for }1\leq i\leq s\right\} = \bigcap_{p} \mathbb{N}^{s} \setminus \left(p^{r}\mathbb{N}\right)^{s}.$$
Since divisibility by different prime powers are independent events, we have
$$\mathrm{pr}^{s}_{t}\left(A\right) = \prod_{p}\mathrm{pr}^{s}_{t}\left(\mathbb{N}^{s}\setminus \left(p^{r}\mathbb{N}\right)^{s}\right) = \prod_{p}\left(1 – \mathrm{pr}^{s}_{t}\left(\left(p^{r}\mathbb{N}\right)^{s}\right)\right) =$$
$$\prod_{p}\left(1-p^{-rst}\right) = \frac{1}{\zeta\left(rst\right)},$$ as required.
QED
Now consider the limit distribution $P_{1}$ by taking $t\to1^{+}$, as in [1]. Then, we get the following familiar result.
Corollary. For integers $r,s\geq 1$ such that $rs\geq 2$, the probability that $s$ integers are relatively $r$-prime in the limit distribution $P_{1}$ is $\displaystyle{\frac{1}{\zeta\left(rs\right)}}$.
Proof.
Since $\zeta\left(z\right)$ is analytic to the right of $\mathfrak{R}\left(z\right) = 1$, we have that $$\lim_{t\to1^{+}}\frac{1}{\zeta\left(rst\right)} = \frac{1}{\zeta\left(rs\right)},$$ as required.
QED
As an extra treat, let’s consider the probability that a random integer is prime! But first, a definition.
Definition. The prime zeta function is $$P\left(z\right) = \sum_{p\text{ prime}}p^{-z}.$$
Theorem. The probability that a random integer is prime is $\displaystyle{\frac{P\left(t\right)}{\zeta\left(t\right)}}$.
In the limit distribution $P_{1}$, a random integer being prime has probability $0$.
Proof.
The probability that a random integer is prime is $$\sum_{p\text{ prime}}\frac{p^{-t}}{\zeta\left(t\right)} = \frac{1}{\zeta\left(t\right)}\sum_{p\text{ prime}}p^{-t}=\frac{P\left(t\right)}{\zeta\left(t\right)}.$$
A well-known result, which I will leave to the reader to prove or find in the literature, is that $\zeta\left(z\right){\sim}\frac{1}{z-1}$ as $z\to 1$. Similarly, it is also known that $P\left(z\right){\sim} \log\frac{1}{\left(z-1\right)}$ as $z\to 1$.
[Hints for proving these are provided after this proof.]
Therefore, $$\lim_{t\to 1^{+}} \frac{P\left(t\right)}{\zeta\left(t\right)} = \lim_{t\to 1^{+}} \left(t-1\right)\log\frac{1}{\left(t-1\right)} = 0,$$ as required.
QED
Hint 1. Note that $\displaystyle{\zeta\left(z\right) = 1 + \int_{1}^{\infty} \frac{\mathrm{d}\lfloor x\rfloor}{x^{z}}}$ and that $\lfloor x\rfloor = x – \left\{x\right\}$, where $\left\{x\right\}$ is the fractional part of $x$. Following this a bit should give the asymptotic for $\zeta\left(z\right)$ as $z\to 1$.
Hint 2. Use the Euler product for $\zeta\left(z\right)$ when looking at $\log\zeta\left(z\right)$, then use Möbius inversion, and finally use the asymptotic for $\zeta\left(z\right)$ as $z\to 1$. With a little work, this should give the asymptotic for $P\left(z\right)$ as $z\to 1$.
References
[1] Solomon W. Golomb, “A Class of Probability Distributions on the Integers.” Journal of Number Theory, Volume 2, Issue 2, May 1970, pp. 189-192.
[2] J. E. Nymann, “On the Probability that $k$ Positive integers are Relatively Prime.” Journal of Number Theory, Volume 4, Issue 5, Oct. 1972, pp. 469-473.