Prime Pair Error Lower Bound Conjecture

Let

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

with $\psi_{2}(N,k)$ and $\mathfrak{S}(k)$ as above. Prime Pair Error Lower Bound Conjecture.

$$E(N)\gg N^{2}(\log N)^{2}.$$

The conjecture gives a lower bound of the expected order for the total prime-pair error. The authors report numerical evidence and state that they have not proved it, although they see no apparent barrier; it remains open.