Zeta Zonk VI

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. Suppose that $X,Y \sim \text{Zeta}(s)$ IID. Define $G$ to be the greatest common divisor (GCD) of $X$ and $Y$. For any positive integer $m$, compute $\mathbb{P}\left( G = m \right)$. Evaluate when $s = 2$ and $m = 3$. Round your answer to the nearest thousandth.

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Note: Solve Zeta Zonk I - V before attempting this question.