The number of Goldbach representations of an integer

Abstract:

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 Hypothesis holds. We prove that \[ \sum _{n=1}^{N} R(n) = \frac {N^{2}}{2} -2 \sum _{\rho } \frac {N^{\rho + 1}}{\rho (\rho + 1)} + \mathcal {O}(N \log ^{3}N), \] 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.