On a numerical upper bound for the extended Goldbach conjecture

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.