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  • $\begingroup$ Getting the twin prime conjecture from some niceness property of zeros of L-functions (even the full GRH) seems out of reach at present, so in terms of recent results it seems that Helfgott's proof of weak Goldbach is a more promising place to look for applications of anomalous zeros; since WGC is known to follow on the GRH we are free to assume a counterexample $\beta$ for a nonconditional proof. From what I gather from skimming the articles this isn't the strategy he adopted though, only a numerical verification of GRH for many values. $\endgroup$ Commented May 19, 2013 at 21:13
  • $\begingroup$ Dear Kalman, As you might know Yitang Zhang has recently shown that there exists infinitely many primes with gaps less than 70 million, for the proof see here: annals.math.princeton.edu/articles/7954 . Many mathematicians have started to think about reducing the 70 million down to a smaller number and eventually to two. In the final attack on the twin prime conjecture, we can assume that Siegel zeros do not exist because of Heath-Brown's result. $\endgroup$ Commented May 23, 2013 at 8:05
  • $\begingroup$ Another comment on the Goldbach conjecture: Vinogradov's first proof of the ternary Goldbach relied on the Siegel-Walfisz theorem, which in turn depends on Siegel's bound for the Siegel zero. As a result the constant C, such that all odd n > C are a sum of three primes was ineffective. In order to make the constant effective you have to use the weaker, but effective analogue of Siegel-Walfish. This produces an enormous constant. In order to bring the constant down you have to play games with the Siegel zero. $\endgroup$ Commented May 23, 2013 at 8:19