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 $s$.
We denote $r(n)$ the number of divisors $d \geqslant 1$ of $n$. Show that the series $\sum_{n=1}^{+\infty} r(n) n^{-s}$ converges and that its sum equals $\zeta(s)^2$.