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In Andrew Granville's paper: REFINEMENTS OF GOLDBACH’S CONJECTURE, AND THE GENERALIZED RIEMANN HYPOTHESIS

He said that:

"we show that if a strong form of Goldbach's conjecture is true then every even integer is the sum of two primes from a rather sparse set of primes. Finally we show that an averaged strong form of Goldbach's conjecture is equivalent to the Generalized Riemann Hypothesis"

Later on, he found that there were some errors in this paper, he wrote another paper: Corrigendum to" Refinements of Goldbach's conjecture, and the Generalized Riemann Hypothesis"

In the second paper, he modified some results of previous paper, but it is not clear to me if his original argument: " an averaged strong form of Goldbach conjecture is equivalent to GRH" still holds ?

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This follows from the answer I posted here.

Define $$G(2N)=\sum_{p+q=2N}\log p\log q$$ to be a weighted count of the number of ways to represent $2N$ as a sum of two primes. Then it is conjectured that $G(2n)\sim J(2N)$ where $$J(2N)=2NC_2 \prod_{p|N,\ p>2}\left(\frac{p-1}{p-2}\right)$$ and $C_2$ is the twin prime constant. The strongest quantitative form of Golbach's conjecture is the statement that $$G(2N)=J(2N)+O_\epsilon(N^{1/2+\epsilon})$$ for every $N$ and arbitrary $\epsilon>0$, and such a result would imply the Riemann hypothesis by the theorem in this answer.

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  • $\begingroup$ how would proving the Goldbach's conjecture ($2N$ is the sum of two primes, $2N>2$) prove $G(2N)=J(2N)+O_\epsilon(N^{1/2+\epsilon})$? $\endgroup$ – onepound Feb 5 '18 at 11:43

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