A Ces\`aro Average of Goldbach numbers

TL;DR

This paper derives an explicit formula for the Cesàro average of Goldbach numbers involving the non-trivial zeros of the Riemann zeta-function, connecting prime representations to zeta zeros.

Contribution

It provides a new explicit formula for the Cesàro average of Goldbach numbers incorporating the zeros of the Riemann zeta-function, advancing understanding of Goldbach representations.

Findings

Explicit formula involving zeta zeros and Goldbach numbers

Asymptotic behavior of the Cesàro average with error term

Connection between Goldbach representations and zeta zeros

Abstract

Let $\Lambda$ 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 $N \geq 2$ 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 $\zeta(s)$.