The number of Goldbach representations of an integer

TL;DR

Under the assumption of the Riemann Hypothesis, the paper derives an explicit formula for the sum of Goldbach representations up to N, linking it to the zeros of the Riemann zeta function and providing an asymptotic estimate.

Contribution

The paper establishes a new explicit formula for the sum of Goldbach representations assuming RH, connecting it to the zeros of the zeta function and advancing understanding of Goldbach's problem.

Findings

Derived an explicit formula for  R(n) sum under RH.

Connected Goldbach representations to non-trivial zeros of  zeta function.

Provided an asymptotic estimate with an explicit error term.

Abstract

We prove the following result: Let $N \geq 2$ and assume the Riemann Hypothesis (RH) holds. Then \[ \sum_{n=1}^{N} R(n) =\frac{N^{2}}{2} -2 \sum_{\rho} \frac{N^{\rho + 1}}{\rho (\rho + 1)} + O(N \log^{3}N), \] where $\rho=1/2+i\gamma$ runs over the non-trivial zeros of the Riemann zeta function $\zeta(s)$.