Is $\zeta(s)\sim\sqrt{\frac{\zeta(4s)}{\zeta(2s)}}\prod\limits_{n=1}^\infty\big(1-\frac{2}{p_n^s+p_n^{-s}}\big)^{-1/2}$?

Question

The Riemann Zeta function, denoted by $\zeta(\cdot)$, is defined by the following equation for $s > 1$ and $p_n$ the $n^\text{th}$ prime number. $$\zeta(s)=\prod_{n=1}^\infty\bigg(1-\frac{1}{p_n^s}\bigg).$$

I came across a recent conjectural proof online that

$$\zeta(s)\sim\sqrt{\frac{\zeta(4s)}{\zeta(2s)}}\prod_{n=1}^\infty\bigg(1-\frac{2}{p_n^s+p_n^{-s}}\bigg)^{-1/2}$$

with RHS $\gtrsim$ LHS and a formula for the radical consisting of... Bernoulli numbers??? I don't know.

I thought I might share this to see what your thoughts are on the online paper. If this conjecture is true, what exactly would it tell us about the relationship between prime numbers and the Riemann hypothesis: much or little?

Here area few examples of $s\in\{2,3,4,5\}$ for which the greater the value of $s$, the more accurate the equation. $$\begin{align} \zeta(2)&\simeq\frac{\pi^2}{\sqrt{105}}\prod_{n=1}^\infty\bigg(1-\frac{2}{p_n^2+p_n^{-2}}\bigg)^{-1/2}\ (0.00127009\%\text{ error})\\ \zeta(3)&\simeq \pi^3\sqrt{\frac{691}{675675}}\prod_{n=1}^\infty\bigg(1-\frac{2}{p_n^3+p_n^{-3}}\bigg)^{-1/2}\ (\text{nearly exact henceforth})\\ \zeta(4)&\simeq \pi^4\sqrt{\frac{3617}{34459425}}\prod_{n=1}^\infty\bigg(1-\frac{2}{p_n^4+p_n^{-4}}\bigg)^{-1/2} \\ \zeta(5)&\simeq \pi^5\sqrt{\frac{174611}{16368226875}}\prod_{n=1}^\infty\bigg(1-\frac{2}{p_n^5+p_n^{-5}}\bigg)^{-1/2}\end{align}$$

What does anyone have to say about this? Is this going to revolutionize the way we perceive the Riemann Zeta function; i.e. is this exciting news for the math community? Can this be entirely confirmed or not? :)

I have no idea, really. Me loving math and all just knows that $\zeta(\cdot)$ is a very important function.

Answer 1

I am sorry, I don’t understand the sense of your question. Because, it's trivial:

Using $~\displaystyle\zeta(s)=\prod\limits_p\frac{1}{1-p^{-s}}~$ we get $~\displaystyle\frac{\zeta(s)^2\zeta(2s)}{\zeta(4s)}=\prod\limits_p\frac{1}{1-\frac{2}{p^s+p^{-s}}}~$

by using $~\displaystyle \frac{1}{1-\frac{2}{x+x^{-1}}}=\frac{x+x^{-1}}{x+x^{-1}-2}=\frac{(\frac{1}{1-x^{-1}})^2\frac{1}{1-x^{-2}}}{\frac{1}{1-x^{-4}}}~$ .

There's nothing special to say about the primes here.

The question is reducible on a discussion of $~\displaystyle\frac{\zeta(2s)}{\zeta(4s)}=\prod\limits_p (1+p^{-2s})~$ .

$\displaystyle n\in\mathbb{N} : ~~\zeta(2n)=(-1)^{n-1}\frac{(2\pi)^{2n}}{2(2n)!}B_{2n}~$ where $~B_k~$ are the Bernoulli numbers .