Zeta Zonk I

We say that $X$ follows a Zeta$(s)$ distribution if it has support on the positive integers with PMF

$$\mathbb{P}[X=k] = \dfrac{k^{-s}}{\zeta(s)}$$

Here, $\zeta(s) = \displaystyle \sum_{k=1}^{\infty} k^{-s}$, and we require $s > 1$. Let $p_1 < p_2 < p_3 < \dots$ be all the prime integers. For positive integers $k$, define $D_k$ to be the event that $p_k$ divides $X$. Find $\mathbb{P}\left( D_k \right)$ when $k = 4$ and $s = 3$.