Some Identities Involving the Cesàro Average of the Goldbach Numbers

Let $\Lambda(n)$ be the von Mangoldt function, and let $r_{G}(n):=\sum_{m_{1}+m_{2}=n}\Lambda(m_{1})\Lambda(m_{2})$ be the weighted sum for the number of Goldbach representations which also includes powers of primes. Let $\widetilde{S}(z):=\sum_{n\geq1}\Lambda(n)e^{-nz}$, where $\Lambda(n)$ is the Von Mangoldt function, with $z\in\mathbb{C}, \mathrm{Re}(z)>0$. In this paper, we prove an explicit formula for $\widetilde{S}(z)$ and the Cesàro average of $r_{G}(n)$.