I came across the following formula due to Ramanujan:


Can someone show me what the proof of this looks like, or point me to a reference (in English)?


1 Answer 1


By Euler's product, for any $s>1$ we have $$ \zeta(s)=\sum_{n\geq 1}\frac{1}{n^s} = \prod_{p}\left(1-\frac{1}{p^s}\right)^{-1} \tag{A}$$ hence $$ \prod_{p}\frac{p^2+1}{p^2-1} = \prod_p\frac{1-\frac{1}{p^4}}{\left(1-\frac{1}{p^2}\right)^2} = \frac{\zeta(2)^2}{\zeta(4)} = \frac{90}{36} = \frac{5}{2}.\tag{B}$$ Similarly, $\prod_{p}\frac{p^4+1}{p^4-1}=\frac{7}{6}$.


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