Riemann hypothesis equivalent to $\prod \left(\frac{4 w_i^2 + 9}{4 w_i^2 + 1}\right) = \prod (1 - 2/v_i) = \frac{\pi}{6}$ ??

Question

Someone told me that the Riemann Hypothesis is equivalent to

$$\prod (1 - 2/v_i) = \frac{\pi}{6}$$

where the product is over all the nontrivial zero's $v_i$. ( for the product we take conjugate pairs of nontrivial zero's ordered by size )

Is that true ?

If true, how to prove it ?

Can this idea be extended to generalized conjectured of dirichlet series ; In particular the largest real part of a nontrivial zero ?

Consider the imaginary parts of the upper nontrivial zero's $w_i$ then the above statements becomes :

$$\prod \left(\frac{4 w_i^2 + 9}{4 w_i^2 + 1}\right) = \frac{\pi}{6}$$

Does this identity have a name ?

Answer 1

Note that Hadamard factorization gives: $2\xi(s)=\prod (1 - s/v_j)$ where the product is on non-trivial zeroes and is conditionally convergent (by grouping conjugate terms together say or taking, as usual, the product in order of $|\Im v_j|$)

Applying it with $s=2$ and noting that $2\xi(2)=\zeta(2)/\pi=\frac{\pi}{6}$ by the celebrated result of Euler solving the so called Basel problem, one gets:

$\prod (1 - 2/v_i) = 2\xi(2)=\frac{\pi}{6}$

so this is a true identity which has nothing to do with the Riemann Hypothesis