18
$\begingroup$

I came across the following formula due to Ramanujan:

$$\prod_{k=1}^{\infty}{\frac{p_k^2+1}{p_k^2-1}}=\frac{5}{2}.$$

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

$\endgroup$
27
$\begingroup$

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, $\prod_{p}\frac{p^4+1}{p^4-1}=\frac{7}{6}$.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.