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This is a natural follow-up after question 3629282.

It is trivial that the irrationality of Brun's constant $B_2\approx1.90216$ implies that there are infinitely many twin primes: $$ B_2 \mbox{ is irrational } ~\Rightarrow~ \mbox{ twin prime conjecture is true.} \tag{1} $$

Interestingly, this answer (now deleted) claimed that something similar is also applicable to the twin prime constant $C_2$: if we can prove the irrationality of the twin prime constant $$ C_2 = \prod_{p > 2} \left(1-\frac{1}{(p-1)^2} \right) = 0.66016\ldots \qquad\mbox{(product over all odd primes } p) $$ then necessarily there are infinitely many twin primes?!

However, the implication $$ C_2 \mbox{ is irrational } ~\Rightarrow~ \mbox{ twin prime conjecture is true (?)} \tag{2} $$ is not at all obvious to me. To put it mildly, $(2)$ is far less obvious than $(1)$ for Brun's constant $B_2$.

Could anyone please sketch the reasoning behind $(2)$ if you do see how it can be done?

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  • $\begingroup$ Is $C_2$ a product only over twin primes? If so, then if there are only finitely-many, that expression is obviously rational. Hence, if by some magic we could prove that expression irrational, then there must be infinitely-many twin primes. (But that constant could conceivably be rational even if there are infinitely-many twin primes... so there's no clear equivalence...) $\endgroup$ – paul garrett Apr 18 at 21:41
  • $\begingroup$ Paul, $C_2$ is a product over all odd primes. $\endgroup$ – Alex Apr 18 at 21:43
  • $\begingroup$ Ah, thanks for the clarification. Then, as in @Joriki's answer, I know of no connection between infinitude of twin primes and any feature of $C_2$. $\endgroup$ – paul garrett Apr 18 at 21:52
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I don't think that's true. This constant merely reflects the conjectured asymptotic density of twin primes. If that conjecture is true, there are infinitely many twin primes, irrespective of whether the constant is rational. And if the conjecture is false, this constant has nothing to do with twin primes. So I don’t see why there should be a connection between the irrationality of this constant and the infinitude of twin primes.

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  • $\begingroup$ Yes, exactly! While it is totally reasonable that the two statements in (2) may be independently true (i.e. $C_2$ may well be irrational AND there may well be infinitely many twins), nevertheless it seems way too difficult (next to impossible) to logically connect them together, to form a rigorous implication (2). $\endgroup$ – Alex Apr 18 at 21:19

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