The number of Goldbach representations of an integer

Abstract: 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_{\rho} \frac{N^{\rho + 1}}{\rho (\rho + 1)} + O(N \log^{3}N), ] where $\rho=1/2+i\gamma$ runs over the non-trivial zeros of the Riemann zeta function $\zeta(s)$.