It seems to me that a proof of $\alpha\_{n}=o(n)$ where the quantity $\alpha\_{n}$ is defined in [About Goldbach's conjecture](https://mathoverflow.net/questions/61842/about-goldbachs-conjecture) together with the main result of <https://kyushu-u.pure.elsevier.com/en/publications/exceptional-zeros-and-the-goldbach-problem> and Terry Tao's answer to this former question of mine: [Does the existence of a Landau-Siegel zero imply the existence of a complex zero off the critical line?](https://mathoverflow.net/questions/404340/does-the-existence-of-a-landau-siegel-zero-imply-the-existence-of-a-complex-zero) would entail that if every large enough even integer is the sum of two primes, then GRH (the analogue of the Riemann Hypothesis for Dirichlet L-functions) holds true.


Can this be established rigorously?



Does asymptotic Goldbach imply GRH?
Granville (2008) proved that a sufficiently strong average error term for the Goldbach-Hardy-Littlewood conjecture is equivalent to RH, and a refinement of it is equivalent to GRH. See MR2357316 and MR2492859. A closely related result was proved by Bhowmik-Ruzsa (2018) and Bhowmik-Halupczok-Matsumoto-Suzuki (2018). See MR3788238 and MR3867327.


Friedlander-Goldston-Iwaniec-Suriajaya (2022) proved that a weak form of the Goldbach-Hardy-Littlewood conjecture rules out exceptional zeros of Dirichel $L$-functions. See MR4356845.


