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Last August I posted this on mathoverflow: https://mathoverflow.net/questions/71856/a-serendipitous-riemann-identity. I show the (slightly revised) equation below:

$$\zeta (3)=\frac{2\pi^4}{315} \prod _{n=1}^{\infty } \left(\frac{1}{(p_n){}^2-p_n}+1\right)$$

Since the constant, $\frac{2\pi^4}{315}$ contains $\pi$, which is known to be transcendental, wouldn't this prove that $\zeta(3)$ is transcendental?

I have calculated the product through the first million primes and Mathematica's Element[product,Rationals] returns True. Also, I built a continued fraction of 18,500,045 elements.

The product converges to http://oeis.org/A082695

A paper that uses the product: http://jtnb.cedram.org/cedram-bin/article/JTNB_2004__16_1_107_0.pdf

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    $\begingroup$ finite product of rationals is rational. $\endgroup$
    – Turbo
    Commented Aug 7, 2021 at 19:41

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No, because an infinite product of rationals is not necessarily rational.

For instance, $$\prod_{n=1}^\infty \left(1-\frac{1}{4n^2}\right)=\frac{2}{\pi}$$

is not rational.

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    $\begingroup$ No, it wouldn't. It doesn't matter where your numbers come from - an infinite product of rational numbers need not be rational. (It might be, but in general it won't. If it is then you need to prove it.) $\endgroup$ Commented Apr 20, 2012 at 8:24
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    $\begingroup$ What Bruno is telling you is that $(\pi)((3/4)(15/16)(35/36)\dots)=2$, from which you would like to conclude that 2 is transcendental. $\endgroup$ Commented Apr 20, 2012 at 12:53
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    $\begingroup$ For the whole product or the partial products? Try it with my product and tell me what it gives you. In any case that's not an acceptable "proof" $\endgroup$ Commented Apr 21, 2012 at 0:00
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    $\begingroup$ @Bruno, you are right. At convergence, your product is not Rational and not Algebraic. At intermediate steps, it is both. $\endgroup$ Commented Apr 21, 2012 at 20:47
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    $\begingroup$ Dear @Rudy: Yes, the property of being rational (or algebraic, or transcendental, or irrational) does not pass over to the limit of a convergent sequence of elements having the property. $\endgroup$ Commented Apr 21, 2012 at 23:57

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