# A Serendipitous Riemann Identity

I was playing around with pronic numbers and got a plus and minus reversed. The number I got was not what I expected, but I looked it up on OEIS.org and found:

$\zeta(6)/\zeta(2)/\zeta(3)$

Reducing and inserting my series we get:

$\zeta(3)=\frac{2\pi^4}{315}\prod_{n}^{\infty }\left(1-\frac{1}{p_{n}-p_{n}^{2}}\right)$

Is this a known identity? If not, should I send it somewhere?

Note: the product appears to converge very slowly.

• Why don't you write $\frac{2\pi^4}{315} \prod_n^\infty{(1-\frac{1}{p_n-p_n^2})}$? – Yaakov Baruch Aug 2 '11 at 4:28
• This is a straightforward exercise to prove. – David Hansen Aug 2 '11 at 4:30
• Infinite products for values of $\zeta$ have been known since the time of Euler, and your identity is the consequence of combining the ones for $\zeta(2)$, $\zeta(3)$, and $\zeta(6)$. So yes, it is a known identity. – Greg Martin Aug 2 '11 at 22:53
• Thanks, Greg. That's all I wanted to know. I'll resume my normal programming. – Fred Kline Aug 3 '11 at 1:31