On a numerical upper bound for the extended Goldbach conjecture

TL;DR

This paper investigates an upper bound related to the Goldbach conjecture, improves approximation methods for the Buchstab function, and refines the computation of Chen's constant, providing experimental insights into the conjecture's bounds.

Contribution

The authors improve the approximation of the Buchstab function and refine the computation of Chen's constant, offering experimental evidence on the bounds related to the Goldbach conjecture.

Findings

Refined approximation of the Buchstab function using Taylor expansions.

Discretized integrals with finer granularity showed negligible improvement to Chen's constant.

Experimental evidence suggests that significant improvements require fundamentally different methods.

Abstract

The Goldbach conjecture states that every even number can be decomposed as the sum of two primes. Let $D(N)$ denote the number of such prime decompositions for an even $N$. It is known that $D(N)$ can be bounded above by $$ D(N) \leq C^* \Theta(N), \quad \Theta(N):= \frac{N}{\log^2 N}\prod_{\substack{p|N p>2}} \left( 1 + \frac{1}{p-2}\right)\prod_{p>2}\left(1-\frac{1}{(p-1)^2}\right) $$ where $C^*$ denotes Chen's constant. It is conjectured that $C^*=2$. In 2004, Wu showed that $C^* \leq 7.8209$. We attempted to replicate his work in computing Chen's constant, and in doing so we provide an improved approximation of the Buchstab function $\omega(u)$, \begin{align*} \omega(u)=1/u, & \quad (1\leq u\leq 2), (u \omega(u))'=\omega(u-1), & \quad (u\geq 2). \end{align*} based on work done by Cheer and Goldston. For each interval $[j,j+1]$, they expressed $\omega(u)$ as a Taylor expansion about $u=j+1$. We expanded about the point $u=j+0.5$, so $\omega(u)$ was never evaluated more than $0.5$ away from the center of the Taylor expansion, which gave much stronger error bounds. Issues arose while using this Taylor expansion to compute the required integrals for Chen's constant, so we proceeded with solving the above differential equation to obtain $\omega(u)$, and then integrating the result. Although the values that were obtained undershot Wu's results, we pressed on and refined Wu's work by discretising his integrals with finer granularity. The improvements to Chen's constant were negligible (as predicted by Wu). This provides experimental evidence, but not a proof, that were Wu's integrals computed on smaller intervals in exact form, the improvement to Chen's constant would be negligible. Thus, any substantial improvement on Chen's constant likely requires a radically different method to what Wu provided.