Some identities involving the Ces\`aro average of Goldbach numbers

TL;DR

This paper derives identities involving the Cesàro average of Goldbach numbers, connecting sums over primes, the Riemann zeta zeros, and special functions, providing explicit formulas and new analytical tools for understanding Goldbach representations.

Contribution

It introduces new identities linking Cesàro averages of Goldbach numbers with the zeros of the Riemann zeta function and explicit formulas involving special functions.

Findings

Derived an explicit identity for the generating function of Goldbach numbers.

Established a formula for the Cesàro average sum involving zeta zeros and special functions.

Provided explicit expressions for sums over Goldbach representations in terms of complex analysis.

Abstract

Let $\Lambda\left(n\right)$ be the von Mangoldt function and $r_{G}\left(n\right) := \sum_{m_1 + m_2=n} \Lambda \left(m_1 \right) \Lambda\left(m_2 \right)$ 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 $\widetilde{S }\left(z\right) := \sum_{n\geq1} \Lambda\left(n\right) e^{-nz}$, with $z\in\mathbb{C}$, $\mathrm{Re}\left(z\right)>0$. In this paper we will prove the identity $$\widetilde{S}\left(z\right) = \frac{e^{-2z}}{z}-\sum_{\rho}z^{-\rho} \Gamma \left(\rho\right) + \sum_{\rho} \left(z^{-\rho} \gamma\left(\rho,2z\right) - \frac{2^{\rho}e^{-z}}{\rho} \right) + G\left(z\right)$$ where $\gamma\left(\rho,2z\right)$ is the lower incomplete Gamma function, $\rho=\beta+i\gamma$ 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_{\rho}\frac{\left(N-2\right)^{\rho+2}}{\rho\left(\rho + 1\right)\left(\rho+2\right)} + & \sum_{\rho_1} \sum_{\rho_2} \frac{\Gamma\left(\rho_{1}\right) \Gamma\left(\rho_{2}\right)} {\Gamma\left(\rho_{1} + \rho_{2}+ 2\right)} N^{\rho_1 + \rho_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.