Prime Pair Error Conjecture

Let

$$E(N)=\sum_{1\leq |k|\leq N}\left(\psi_{2}(N,k)-\mathfrak{S}(k)(N-|k|)\right)^{2},$$

where $\psi_{2}(N,k)$ counts the von Mangoldt-weighted prime pairs of difference $k$, and $\mathfrak{S}(k)$ is the Hardy–Littlewood singular series. Prime Pair Error Conjecture. There exists a positive constant $c_{3}$ such that

$$E(N)\sim c_{3}N^{2}(\log N)^{2}\quad\text{as }N\to\infty.$$

The paper states that this conjecture implies the Riemann Hypothesis, while even the required upper bounds for $E(N)$ are difficult; it remains open.