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My question pertains to two famous groups of related conjectures:

Goldbach's Conjecture (GC); Goldbach's Weak Conjecture (GWC); The Riemann Hypothesis (RH); The Generalized Riemann Hypothesis (GRH).

These two groups of conjectures both appear to be strong statements about the distribution of the prime numbers. Yet it is unclear to me if each provides independent information about the distribution of the primes.

If the two conjectures expressed the same underlying fact, we would expect a strong coupling between the conjectures, such as GC if and only if RH. Therefore, I am asking if someone would someone please take the time to review the state of the reseach on the relationships between GC, GWC, RH, and GRH. References for any results mentioned would be much appreciated.

For my part, I've read that by assuming GRH, one can provide an asymptotic proof of GWC. This strikes me as a weak relationship, and I wonder if there are stronger results which make use of RH to prove GC.

In the other direction, I've heard one person claim, without references, that GC implies RH. The latter claim seems unlikely to me, but I'd like to find someone that can set the record straight.

Note: I am looking for relationships across the two conjectures. It is clear to me that GC implies GWC, and GRH implies RH.

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    $\begingroup$ A related question. Also here, page $8$, final paragraph. This might also prove helpful. $\endgroup$ – Lucian Feb 11 '14 at 1:27
  • $\begingroup$ Thank you. I've seen the first post, and I'll take a look at the second. Just to keep information in one place, an outline of an asymptotic proof of GWC assuming GRH can be found here. This reference was provided by one one of the posters from Lucian's first link. $\endgroup$ – Doug Feb 11 '14 at 1:35
  • $\begingroup$ The link I provided in the comment below outlines a proof strategy to "show that an averaged strong form of Goldbach’s conjecture is equivalent to the Generalized Riemann Hypothesis". It is the same link as Lucian's third link. $\endgroup$ – Doug Feb 11 '14 at 22:33
  • $\begingroup$ Lucian's second link provides a reference that RH implies GH asymptotically: "In 1997, Deshouillers, Enger, te Riele and Zinoviev proved that the generalized Riemann hypothesis implies Goldbach's strong conjecture [Borwein et al., 2008]" $\endgroup$ – Doug Feb 11 '14 at 22:36
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Here is a related question that addresses your concerns(see the answer):

https://mathoverflow.net/questions/61842/about-goldbachs-conjecture/62109#62109

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  • $\begingroup$ From that discussion, here is a reference to a paper that claims to "show that an averaged strong form of Goldbach’s conjecture is equivalent to the Generalized Riemann Hypothesis". It is not yet clear to me what an average strong form of GC is. $\endgroup$ – Doug Feb 11 '14 at 1:46

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