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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 ?

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  • $\begingroup$ The generalized Riemann hypothesis implies Goldbach's weak conjecture. The other claim need not be true. $\endgroup$ – Dietrich Burde Aug 29 '15 at 17:39
  • $\begingroup$ So, the Riemann hypothesis does NOT necessarily imply the (Strong) Goldbach conjecture ? $\endgroup$ – Peter Aug 29 '15 at 17:41
  • $\begingroup$ No, RH for the zeta function alone is not enough. $\endgroup$ – Dietrich Burde Aug 29 '15 at 17:45
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    $\begingroup$ This link : primes.utm.edu/glossary/xpage/goldbachconjecture.html claims that the Riemann-hypothesis (not generalized) implies the ODD Goldbach conjecture. Is this true, or do we need the generalized version even for the weak Goldbach conjecture ? $\endgroup$ – Peter Aug 29 '15 at 18:09
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    $\begingroup$ @Peter: Odd Goldbach has been a theorem of Helfgott since 2013. Unconditionally (not depending on RH or GRH). $\endgroup$ – Jyrki Lahtonen Aug 29 '15 at 18:29
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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.

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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."

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  • $\begingroup$ Here's a problem: Suppose someone using techniques not remotely related to Riemann's $\zeta$ function proves the twin prime conjecture. Then the statement "If R.H. then twinprimeconjecture" is (vacuously) true. Can one then PROVE that in some sense R.H. does not imply the twin prime conjecture? Maybe one could speak of some nonstandard models in which R.H. is true and the twin prime conjecture is false, and that would show the impossibility of certain kinds of proofs that T.P.C. follows from R.H. I suspect that's not easy. ${}\qquad{}$ $\endgroup$ – Michael Hardy Aug 29 '15 at 19:32

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