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Can anyone help me with the following implication of the GRH which is fundamental while proving the Odd Goldbach's Conjecture?

$$\psi(x,\chi):=\sum_{n\leq x}\Lambda(n)\chi(n)=O(x^{1/2}\log^2 x)$$ where $\chi$ is a non-trivial Dirichlet character.

Also tell me the difference in proof while considering the trivial Dirichlet Character, $\chi$.

Thank You.

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It is not clear what you mean by "fundamental while proving the Odd Goldbach's Conjecture", since the "conjecture" was proved by Vinogradov without GRH. – GH from MO Oct 18 '12 at 12:57
Vinogradov's proof showed that every sufficiently large odd integer can be written as the sum of three primes; concluding this for every odd integer greater than 6 requires GRH (so far). – Greg Martin Oct 20 '12 at 3:26
@Greg: Thanks for the clarification! – GH from MO Oct 20 '12 at 16:46
@ GH: by "fundamental while proving the Odd Goldbach's Conjecture", I mean the proof of the vinogradov's theorem when it is proved by assuming the GRH. Of course in the original proof of Vinogradov, he did not use anything of GRH. I was actually stuck with the formulae while studying the proof of it under the GRH. @Greg: I didn't know actually that extending vinogradov's theorem for every odd numbers requires GRH. Thanks again but I will check that out. – Ankush Oct 22 '12 at 11:45

This can be found, for example, as Theorem 13.7 in Montgomery and Vaughan's Multiplicative Number Theory I. Classical Theory.

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That's really helpful. Thanks a lot. – Ankush Oct 18 '12 at 12:45

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