Let $x$ be a real number such that $x > 1$ and let $X$ be a random variable that follows the zeta distribution with parameter $x$, i.e. $$\forall n \in \mathbb{N}^{*}, \quad \mathbb{P}(X = n) = \frac{1}{\zeta(x) n^{x}}$$ For all $n \in \mathbb{N}^{*}$, denote by $B_{n}$ the event $B_{n} = \bigcap_{k=1}^{n} (X \notin p_{k}\mathbb{N}^{*})$, where $p_1 < p_2 < \cdots$ are the prime numbers in increasing order. Show that $\lim_{n \rightarrow \infty} \mathbb{P}(B_{n}) = \mathbb{P}(X = 1)$. Deduce that $$\forall x \in {]1,+\infty[}, \quad \frac{1}{\zeta(x)} = \lim_{n \rightarrow +\infty} \prod_{k=1}^{n} \left(1 - \frac{1}{p_{k}^{x}}\right)$$
Let $x$ be a real number such that $x > 1$ and let $X$ be a random variable that follows the zeta distribution with parameter $x$, i.e.
$$\forall n \in \mathbb{N}^{*}, \quad \mathbb{P}(X = n) = \frac{1}{\zeta(x) n^{x}}$$
For all $n \in \mathbb{N}^{*}$, denote by $B_{n}$ the event $B_{n} = \bigcap_{k=1}^{n} (X \notin p_{k}\mathbb{N}^{*})$, where $p_1 < p_2 < \cdots$ are the prime numbers in increasing order.
Show that $\lim_{n \rightarrow \infty} \mathbb{P}(B_{n}) = \mathbb{P}(X = 1)$. Deduce that
$$\forall x \in {]1,+\infty[}, \quad \frac{1}{\zeta(x)} = \lim_{n \rightarrow +\infty} \prod_{k=1}^{n} \left(1 - \frac{1}{p_{k}^{x}}\right)$$