# Would Elliott-Halberstam conjecture follow from GRH?

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?
Edit March 29th 2021: denoting by $$\Theta_{EH}:=\sup\{\theta\vert EH(\theta)\}$$, EH conjecture is equivalent to $$\Theta_{EH}=1$$. The proven lower bound for it has increased from $$1/2$$ (Bombieri-Vinogradov) to a greater value. On the other hand, the proven upper bound for the de Bruijn-Newman constant $$\Lambda$$ has decreased from $$1/2$$ to $$0.2$$ (Platt and Trudgian), while the proof of Newman's conjecture by Rodgers and Tao implies $$\Lambda\geq 0$$, and RH is equivalent to this constant vanishing. So that, it seems that $$\Theta_{EH}$$ and $$\Lambda$$ are "dual", the duality relation being the involution $$s\mapsto 1-s$$. Is it thus likely that $$\theta<1-\Lambda\Longrightarrow EH(\theta)$$?
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).
• EH even implies that $16$ can be replaced by $12$, as shown by James Maynard. – Sylvain JULIEN Dec 29 '13 at 19:17