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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?

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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
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Thank you for the comments. –  joro Apr 13 '13 at 5:37

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