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It seems to me that a proof of $\alpha_{n}=o(n)$ where the quantity $\alpha_{n}$ is defined in About Goldbach's 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? 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?

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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.

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