Let $x \in \mathbb{R}$ such that $x > 1$. Let $X$ and $Y$ be two independent random variables each following a zeta probability distribution with parameter $x$, i.e. $$\forall n \in \mathbb{N}^{*}, \quad \mathbb{P}(X = n) = \mathbb{P}(Y = n) = \frac{1}{\zeta(x) n^{x}}$$ Let $A$ be the event ``No prime number divides $X$ and $Y$ simultaneously''. For all $n \in \mathbb{N}^{*}$, denote by $C_{n}$ the event $$C_{n} = \bigcap_{k=1}^{n} \left((X \notin p_{k}\mathbb{N}^{*}) \cup (Y \notin p_{k}\mathbb{N}^{*})\right)$$ Express the event $A$ using the events $C_{n}$. Deduce that $$\mathbb{P}(A) = \frac{1}{\zeta(2x)}$$
Let $x \in \mathbb{R}$ such that $x > 1$. Let $X$ and $Y$ be two independent random variables each following a zeta probability distribution with parameter $x$, i.e.
$$\forall n \in \mathbb{N}^{*}, \quad \mathbb{P}(X = n) = \mathbb{P}(Y = n) = \frac{1}{\zeta(x) n^{x}}$$
Let $A$ be the event ``No prime number divides $X$ and $Y$ simultaneously''. For all $n \in \mathbb{N}^{*}$, denote by $C_{n}$ the event
$$C_{n} = \bigcap_{k=1}^{n} \left((X \notin p_{k}\mathbb{N}^{*}) \cup (Y \notin p_{k}\mathbb{N}^{*})\right)$$
Express the event $A$ using the events $C_{n}$. Deduce that
$$\mathbb{P}(A) = \frac{1}{\zeta(2x)}$$