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 for all $k \in \mathbb{N}^*$, the random variable $\nu_{p_k}(X) + 1$ follows the geometric distribution with parameter $\left(1 - p_k^{-s}\right)$.