MathOverflow is a question and answer site for professional mathematicians. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

For $k \in \mathbb{N}$ define $$ f(k) = \prod_{\text{p prime}}\left(1+\frac{1}{p^k(p^k+1)}\right)$$ $$ g(k) = \prod_{\text{p prime}}\left(1+\frac{1}{p^k(p^k-1)}\right)$$

By the product for zeta $f(1)=\zeta(2)/\zeta(3)$.

With 100 digits of precision and Cohen's pari script $f(2)=21/(2\pi^2)$ ,$g(1)=315/2\zeta(3)/\pi^4$.

Are the conjectured values correct?

Other closed forms?

share|cite|improve this question
Dear joro, The same argument that computes $f(1)$ shows that $f(k) = \zeta(2k)/\zeta(3k)$. This explains the value of $f(2)$. A similar argument shows that $g(k) = \zeta(2k)\zeta(3k)/\zeta(6k)$, which explains the value of $g(1)$. Regards, – Emerton Apr 12 '13 at 14:51
Thank you for the comments. – joro Apr 13 '13 at 5:37

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Browse other questions tagged or ask your own question.