@Article{LanguascoZaccagnini2012, Author = "A. Languasco and A. Zaccagnini", Title = "The number of {Goldbach} representations of an integer", Journal = "Proc. Amer. Math. Soc.", Volume = 140, Issue = 3, Pages = "795--804", Year = 2012 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 $N \ge 2$ and assume that the Riemann Hypothesis holds. We prove that \[ \sum_{n = 1}^N R(n) = \frac{N^{2}}{2} - 2 \sum_{\rho} \frac{N^{\rho + 1}}{\rho (\rho + 1)} + \Odi{N (\log N)^3}, \] where $\rho = 1/2 + i \gamma$ runs over the non-trivial zeros of the Riemann zeta-function $\zeta(s)$." }