MathWorld's page Prime Products gives the 'related result' (7) to Mertens' theorem: $$ \lim_{n\to\infty}\log p_n\prod_{k=1}^n\frac{1}{1+1/p_k}=\frac{\pi^2}{6e^\gamma}. $$

Does this identity have a classical reference? When possible I like to give credit to original discoverers rather than merely citing it as 'easy to show'.


Mertens' Theorem says:

$$\lim_{n \rightarrow \infty} \ \frac{1}{\log p_n} \prod_{k=1}^{n} \frac{1}{1 - \displaystyle{\frac{1}{p_k}}} = e^{\gamma}.$$

Euler's product formula for the $\zeta$ function and his evaluation of $\zeta(2) = \pi^2/6$ says that

$$\zeta(2) = \lim_{n \rightarrow \infty} \ \prod_{k=1}^{n} \frac{1}{1 - \displaystyle{\frac{1}{p^2_k}}} = \lim_{n \rightarrow \infty} \ \prod_{k=1}^{n} \frac{1}{\left(1 - \displaystyle{\frac{1}{p_k}}\right)\left(1 + \displaystyle{\frac{1}{p_k}}\right)} = \frac{\pi^2}{6}.$$

Your result is the second limit divided by the first. The derivation of this is so immediate that I doubt you will find it as a "result" in the literature. The correct people to cite are Mertens and Euler.


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