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The Wikipedia article about Elliott-Halberstam (EH for short) conjecture says that the so-called Bombieri-Vinogradov theorem, which is a weaker form of EH conjecture, is in some sense an averaged form of the Generalized Riemann Hypothesis (that is, the analogue of RH for Dirichlet L-functions). So, would the full EH conjecture follow from GRH?
Thanks in advance.

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The Elliott-Halberstam conjecture is not known to follow from GRH. Even the weak version of EH (which is with $Q=x^{1/2+\epsilon}$ for any fixed $\epsilon>0$) does not follow from GRH. On the other hand, it is known that the Elliott-Halberstam conjecture almost implies the twin primes conjecture, i.e., it implies that there are infinitely many pairs of primes at distance $≤ 16$ (now $\le 12$, see Sylvain's comment). Furthermore, the Bombieri-Vinogradov theorem is indeed an amazingly strong unconditional replacement for the GRH bound (and has as natural strengthening the EH conjecture).

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EH even implies that $16$ can be replaced by $12$, as shown by James Maynard. –  Sylvain JULIEN Dec 29 '13 at 19:17
    
I think a big problem in preventing GRH from showing EH is the possible non-uniformity in implicit constants in the error terms. Perhaps a uniform version of GRH would suffice. –  Stanley Yao Xiao Dec 30 '13 at 0:59

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