The weakened Goldbach upper-bound conjecture

Let $\Lambda$ be the von Mangoldt function, let

$$\psi_2(n)=\sum_{m+m'=n}\Lambda(m)\Lambda(m'),$$

and let $\mathfrak{S}(n)$ be the singular series defined by

$$\mathfrak{S}(n)=2C_2\prod_{\substack{p\mid n\p>2}}\left(\frac{p-1}{p-2}\right)$$

for even $n$, with $C_2=\prod_{p>2}(1-(p-1)^{-2})$. The weakened Goldbach upper-bound conjecture. Given a fixed constant $0<\delta<1$, for sufficiently large even $n$,

$$\left|\psi_2(n)-\mathfrak{S}(n)n\right|\leq(1-\delta)\mathfrak{S}(n)n.$$

Equivalently,

$$\delta\mathfrak{S}(n)n\leq\psi_2(n)\leq(2-\delta)\mathfrak{S}(n)n.$$

This is a weaker form of the Hardy–Littlewood Goldbach conjecture, retaining both a positive lower bound and an upper bound strictly below twice the conjectured main term. The paper uses it as a hypothesis; its general validity remains open.