Does the Riemann-Hypothesis imply the Twin-Prime-Conjecture?

Question

The Riemann hypothesis (https://en.wikipedia.org/wiki/Riemann_hypothesis) is one of the most important conjectures in number theory. I read that the Riemann hypothesis implies the Goldbach Conjecture and would allow much better estimates for the prime-counting function.

What about the Twin-Prime-Conjecture ?

Would it follow from the Riemann-hypothesis ?

Answer 1

I'd say no. Many authors state that RH would tell us nothing more (about prime gaps) than $p_{n+1}-p_n \in \text{O}(\sqrt{p_n}\log p_n)$, which obviously doesn't imply TPC, and so it should not be unsafe to say RH doesn't imply TPC.

Answer 2

I think that RH does not imply the twin prime conjecture. A couple of quotations from Dan Goldston in his paper here are in favour of this opinion:

"While the Riemann Hypothesis is decisive in determining the distribution of primes, it seems to be of little help with regard to twin primes."

"The conjecture that the distribution of twin primes satisfies a Riemann Hypothesis type error term is well supported empirically, but I think this might be a problem that survives the current millennium."

Answer 3

Interestingly, as I recently learned from a popular Mathoverflow question, if the RH GRH is false in a specific way (it has Siegel zeros), that would imply the Twin Prime conjecture is true (!!)