Strong twisted pair correlation conjecture

Let $n$ be a prime power, define

$$F_n(\alpha)=-\left(\frac{T}{2\pi}\frac{\Lambda(n)}{\sqrt n}\right)^{-1}\sum_{T\leq\gamma,\gamma'\leq2T}n^{i\gamma}T^{i\alpha(\gamma-\gamma')}\omega(\gamma-\gamma'),$$

and define

$$r_1(\alpha,n)=\frac1{\Lambda(n)}\sum_m\frac{\Lambda(mn)\Lambda(m)}m\min\left\{\frac m{T^\alpha},\frac{T^\alpha}m\right\}^2,$$ $$r_2(\alpha,n)=\frac1{\Lambda(n)}\sum_m\Lambda(m)\Lambda(n/m)\min\left\{\frac{nT^\alpha}m,\frac m{nT^\alpha}\right\}^2.$$

Strong twisted pair correlation conjecture. Fix $\varepsilon>0$. Uniformly for all prime powers $n\leq T^{1-\varepsilon}$,

$$F_n(\alpha)=\begin{cases}T^{2\alpha}(\log T+O(1))+\dfrac{\log T+O(1)}{(nT^\alpha)^2}-r_2(\alpha,n)+O(1/\log T),&-\dfrac{\log n}{\log T}\leq\alpha\leq0,\\ T^{-2\alpha}(\log T+\dfrac{\log T}{n^2}+O(1))-r_1(\alpha,n)+O(1/\log T),&0<\alpha<1-\dfrac{\log n}{\log T},\\ \min\{1,\dfrac{\log T}{\Lambda(n)}(\alpha-1+\dfrac{\log n}{\log T})\}+O(1/\log T),&\alpha\geq1-\dfrac{\log n}{\log T}. \end{cases}$$

This conjecture gives the detailed asymptotic for the twisted pair-correlation function needed in the paper's third-moment analysis. The source explains the delta-like behavior of the first terms and the localized nature of the $r_1,r_2$ corrections; it proves restricted ranges but leaves the full uniform statement open.