Divergent Products equal to π

List of divergent infinite products whose value is $\pi$ .

Product over ℕ

$\displaystyle \prod_{n\geq1} \dfrac{1}{\pi^2} = \pi$

$\displaystyle \prod_{n\geq1} 4n^2 = \displaystyle \prod_{n\geq1} (2n)^2 = \pi$

$\displaystyle \prod_{n\geq2} (n^2-1) = \pi$

$\displaystyle \prod_{n\geq1} \dfrac{n^2}{4n^2-1} = \pi$

$\displaystyle \prod_{n\geq1} \dfrac{1}{16} \left(1-\dfrac{1}{(2n+1)^2}\right) = \pi$

Product over ℙ

$\displaystyle \prod_{p\in\mathbb{P}} 4 \sqrt{p} = \pi$

$\displaystyle \prod_{p\in\mathbb{P}} 96 \sqrt{p^2-1} = \pi$

$\displaystyle \prod_{p\in\mathbb{P}} 240 \sqrt{p^2+1} = \pi$

$\displaystyle \prod_{p\in\mathbb{P}} 48\sqrt{10} \, \sqrt[4]{p^4-1} = \pi$

$\displaystyle \prod_{p\in\mathbb{P}} \dfrac{\sqrt{6}\,p}{\sqrt{p^2-1}} = \pi$