The Wikipedia article about ElliottHalberstam (EH for short) conjecture says that the socalled BombieriVinogradov 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 Lfunctions). So, would the full EH conjecture follow from GRH?
Thanks in advance.
The ElliottHalberstam 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 ElliottHalberstam 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 BombieriVinogradov theorem is indeed an amazingly strong unconditional replacement for the GRH bound (and has as natural strengthening the EH conjecture).

2$\begingroup$ EH even implies that $16$ can be replaced by $12$, as shown by James Maynard. $\endgroup$ – Sylvain JULIEN Dec 29 '13 at 19:17

1$\begingroup$ I think a big problem in preventing GRH from showing EH is the possible nonuniformity in implicit constants in the error terms. Perhaps a uniform version of GRH would suffice. $\endgroup$ – Stanley Yao Xiao Dec 30 '13 at 0:59