Some identities involving the Cesàro average of Goldbach numbers

Abstract

Let $Λ\left(n\right)$ be the von Mangoldt function and $r_{G}\left(n\right) := \sum_{m_1 + m_2=n} Λ\left(m_1 \right) Λ\left(m_2 \right)$ be the counting function for the numbers that can be written as sum of two primes (that we will call "Goldbach numbers", for brevity) and let $\widetilde{S }\left(z\right) := \sum_{n\geq1} Λ\left(n\right) e^{-nz}$, with $z\in\mathbb{C}$, $\mathrm{Re}\left(z\right)>0$. In this paper we will prove the identity $$\widetilde{S}\left(z\right) = \frac{e^{-2z}}{z}-\sum_ρz^{-ρ} Γ\left(ρ\right) + \sum_ρ \left(z^{-ρ} γ\left(ρ,2z\right) - \frac{2^ρe^{-z}}ρ \right) + G\left(z\right)$$ where $γ\left(ρ,2z\right)$ is the lower incomplete Gamma function, $ρ=β+iγ$ runs over the non-trivial zeros of the Riemann Zeta function and $G\left(z\right)$ is a sum of (explicitly calculate) elementary function and complex Exponential integrals. In addition we will prove that \begin{align*} \sum_{n\leq N} r_G \left(n\right) \left(N-n\right) = & \frac{N^{3}}{6} - 2\sum_ρ\frac{\left(N-2\right)^{ρ+2}}{ρ\left(ρ+ 1\right)\left(ρ+2\right)} + & \sum_{ρ_1} \sum_{ρ_2} \frac{Γ\left(ρ_{1}\right) Γ\left(ρ_{2}\right)} {Γ\left(ρ_{1} + ρ_{2}+ 2\right)} N^{ρ_1 + ρ_2+1} + F\left(N\right) \end{align*} where $N>4$ is a natural number and $F\left(N\right)$ is a sum of (explicitly calculate) elementary functions, dilogarithms and sums over non-trivial zeros of the Riemann Zeta function involving the incomplete Beta function.