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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:

$\frac{\pi}{4}=\frac34\cdot\frac54\cdot\frac78\cdot\frac{11}{12}\cdot\frac{13}{12}\cdots$

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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    @J.M. I know. I just thought it had something similar to the above.2012-08-05

1 Answers 1

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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.