The number of Goldbach representations of an integer

Alessandro Languasco, Alessandro Zaccagnini

We prove the following result: Let $N\ge 2$ 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 \l…