The number of Goldbach representations of an integer

We prove the following result: Let $N \geq 2$ and assume the Riemann Hypothesis (RH) holds. Then $\sum_{n=1}^{N} R(n) =\frac{N^{2}}{2} -2 \sum_ρ \frac{N^{ρ+ 1}}{ρ(ρ+ 1)} + O(N \log^{3}N),$ where $ρ=1/2+iγ$ runs over the non-trivial zeros of the Riemann zeta function $ζ(s)$.

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