A Ces\`aro Average of Goldbach numbers

Let $\mathrm{\Lambda }$ be the von Mangoldt function and $(r_G(n) = \sum_{m_1 + m_2 = n} \Lambda(m_1) \Lambda(m_2))$ be the counting function for the Goldbach numbers. Let $N\ge 2$ be an integer. We prove that $$\begin{align} &\sum_{n \le N} r_G(n) \frac{(1 - n/N)^k}{\Gamma(k + 1)} = \frac{N^2}{\Gamma(k + 3)} - 2 \sum_\rho \frac{\Gamma(\rho)}{\Gamma(\rho + k + 2)} N^{\rho+1}\ &\qquad+ \sum_{\rho_1} \sum_{\rho_2} \frac{\Gamma(\rho_1) \Gamma(\rho_2)}{\Gamma(\rho_1

• \rho_2 + k + 1)} N^{\rho_1 + \rho_2} + $\mathcal{O}$_k(N^{1/2}), \end{align}$$for$k > 1$,where$\rho$, with or without subscripts, runs over the non-trivial zeros of the Riemann zeta-function $\zeta (s)$.