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 if $p$ is a prime number such that $p \equiv 1 [4]$, we have
$$E\left(g\left(p^{\nu_p(X)}\right)\right) = \frac{1}{1 - p^{-s}}.$$