For every integer $k \geqslant 2$, we set $\zeta(k) = \sum_{n=1}^{+\infty} n^{-k}$. For fixed $s > 1$, we define a probability distribution $\mu_s$ on $\mathbb{N}^*$ by setting, for $n \in \mathbb{N}^*$, $$\mu_s(\{n\}) = \frac{1}{\zeta(s) n^s}$$ Let $s \geqslant 2$ be an integer. Let $Z_n^{(s)}$ be the gcd of $s$ mutually independent random variables all following the uniform distribution on $\{1, 2, \ldots, n\}$. Using the results of questions 18, 19, and 20a, deduce that the sequence $(\mu_{Z_n^{(s)}})_{n \in \mathbb{N}}$ converges in $\mathscr{B}(\mathscr{P}(\mathbb{N}^*), \mathbb{R})$ to $\mu_s$.
For every integer $k \geqslant 2$, we set $\zeta(k) = \sum_{n=1}^{+\infty} n^{-k}$. For fixed $s > 1$, we define a probability distribution $\mu_s$ on $\mathbb{N}^*$ by setting, for $n \in \mathbb{N}^*$,
$$\mu_s(\{n\}) = \frac{1}{\zeta(s) n^s}$$
Let $s \geqslant 2$ be an integer. Let $Z_n^{(s)}$ be the gcd of $s$ mutually independent random variables all following the uniform distribution on $\{1, 2, \ldots, n\}$. Using the results of questions 18, 19, and 20a, deduce that the sequence $(\mu_{Z_n^{(s)}})_{n \in \mathbb{N}}$ converges in $\mathscr{B}(\mathscr{P}(\mathbb{N}^*), \mathbb{R})$ to $\mu_s$.