Zeta Zonk VII

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. Let $N$ be the event that $X$ is $n-$free (i.e. every integer that is the $n$th power of some other integer that isn't $1$ does not divide $X$). Find $\mathbb{P}[N]$ when $s = 1$ and $n = 4$. Return your answer to the nearest thousandth.

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Note: Complete Zeta Zonk I - VI before attempting this question.