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 recall that $\chi_4(2n) = 0$ and $\chi_4(2n-1) = (-1)^{n-1}$ for $n \in \mathbb{N}^*$.
Show that, if $p$ is a prime number,
$$E\left(\chi_4\left(p^{\nu_p(X)}\right)\right) = \frac{1 - p^{-s}}{1 - \chi_4(p) p^{-s}}.$$