Twisted pair correlation conjecture

Let $n=q^a$ be a prime power, define $\widetilde\gamma=(\log T)/(2\pi)\,\gamma$, and let $\mathfrak C(\mathbb R)$ be the set of continuous $L^1(\mathbb R)$ functions $r$ whose Fourier transform $\hat r$ is integrable and Lipschitz continuous with $\hat r'(a)\ll|a|^{-3}$. Let $\mathcal E_n=(a-1)\log q/\log T+1/\log T$, and define $m_n$ by

$$m_n(\alpha)=\begin{cases}1,&\alpha<-1-\Lambda(n)/\log T,\\(\log T/\Lambda(n))(-\alpha-1),&-1-\Lambda(n)/\log T\leq\alpha<-1,\\0,&-1\leq\alpha<1-\log n/\log T,\\(\log T/\Lambda(n))(\alpha-1+\log n/\log T),&1-\log n/\log T\leq\alpha<1-(\log n-\Lambda(n))/\log T,\\1,&\alpha\geq1-(\log n-\Lambda(n))/\log T. \end{cases}$$

Twisted pair correlation conjecture. Fix $\varepsilon>0$ and let $r\in\mathfrak C(\mathbb R)$. Uniformly for all prime powers $n=q^a\leq T^{1-\varepsilon}$,

$$-\left(\frac{T}{2\pi}\frac{\Lambda(n)}{\sqrt n}\right)^{-1}\sum_{T\leq\gamma,\gamma'\leq2T}n^{i\gamma}r(\widetilde\gamma-\widetilde\gamma')=\int_{\mathbb R}\frac{\hat r(\alpha)+\hat r(-\alpha-\log n/\log T)}2\bigl(\delta(\alpha)+m_n(\alpha)\bigr)\,d\alpha+O(\mathcal E_n).$$

This weaker, integrated formulation follows from the strong twisted pair correlation conjecture by convolution with a suitable kernel. It is introduced because it is the form needed for the later argument; the source supports it conditionally from the strong conjecture but does not establish it in full.