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According to the section Proof using the irrationality of $\pi$ of the Wikipedia article on Euclid's theorem, Euler proved that:


where "each denominator is the multiple of four nearest to the numerator". Can someone please explain this formula? I see it, but cannot believe it.

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A very nice explanation is given here: – anthus Aug 5 '12 at 3:18
$$\frac{\pi^2}{6}=\frac11+\frac14+\frac19+\frac{1}{16}+\frac{1}{25}+\cdots=\prod‌​_{p} \frac{1}{1-p^{-2}}$$ – Makoto Kato Aug 5 '12 at 6:22
@Makoto, it's related, but rather different. – J. M. Aug 5 '12 at 9:59
@J.M. I know. I just thought it had something similar to the above. – Makoto Kato Aug 5 '12 at 10:31

The formula is described here (I am having a hard time finding a more authoritative reference); briefly, the OP's product in the usual product notation goes like

$$\frac{\pi}{4}=\prod_{k=2}^\infty \frac{p_k}{p_k-\chi(p_k)}$$

where $p_k$ is the $k$-th prime and $\chi(n)$ is a character defined as

$$\chi(n)=\begin{cases}1&\text{if }n\equiv 1\pmod 4\\-1&\text{if }n\equiv 3\pmod 4\end{cases}$$

As noted, the derivation is done by treating the usual Leibniz series

$$\frac{\pi}{4}=\sum_{k=1}^\infty \frac{(-1)^{k-1}}{2k-1}$$

as a Dirichlet series, and then expanding that series as an Euler product.

Edit: Daniel has given another nice link in the comments.

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