Let $s > 1$ be a real number and let $X$ be a random variable taking values in $\mathbb{N}^*$ following the zeta distribution with parameter $s$.
Show that the sequence of functions $\left(x \mapsto \prod_{k=1}^{n} p_k^{\nu_{p_k}(x)}\right)_{n \geqslant 1}$ from $\mathbb{N}^*$ to $\mathbb{N}^*$ converges pointwise to the identity function.