Bounds of weighted sums of Mangoldt function under the Riemann Hypothesis

Hello,

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