The number of Goldbach representations of an integer

Let $\Lambda$ be the von Mangoldt function and \(R(n) = \sum_{h+k=n} \Lambda(h)\Lambda(k) \) be the counting function for the Goldbach numbers. Let $N \geq 2$ and assume that the Riemann Hypothdfesis holds. We prove that \[ \sum_{n=1}^{N} R(n) = \frac{N^{2}}{2} -2 \sum_{\rho} \frac{N^{\rho + 1}}{\rho (\rho + 1)} + \Odi{N \log^{3}N }, \] where $\rho=1/2+i\gamma$ runs over the non-trivial zeros of the Riemann Zeta-function $\zeta(s)$. This improves a recent result by Bhowmik and Schlage-Puchta.