The weakened prime-pair upper-bound conjecture

For $k\geq0$, define

$$\psi_2(x,k)=\sum_{n\leq x}\Lambda(n)\Lambda(n-k),$$

where $\Lambda$ is the von Mangoldt function and $\mathfrak{S}(k)$ is the singular series. The weakened prime-pair upper-bound conjecture. Given a fixed constant $0<\delta<1$, for even $2\leq k\leq x$ and sufficiently large $x$,

$$\psi_2(x,k)\leq(2-\delta)\mathfrak{S}(k)(x-k)+o(\mathfrak{S}(k)x).$$

This is an upper-bound weakening of the Hardy–Littlewood prime-pair conjecture, requiring only a bound strictly below twice the conjectured main term. Its validity in the full stated range remains open.