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 $s$. If $n \in \mathbb{N}^*$, we set $g(n) = r_1(n) - r_3(n)$ where $r_i(n) = \operatorname{Card}\{d \in \mathbb{N} : d \equiv i [4] \text{ and } d \mid n\}$.
Show that $E(g(X)) = \lim_{n \rightarrow +\infty} \prod_{k=1}^{n} E\left(g\left(p_k^{\nu_{p_k}(X)}\right)\right)$.