Montgomery's pair correlation conjecture

Let $\zeta$ denote the Riemann zeta function, and assume the Riemann hypothesis, so that every non-trivial zero of $\zeta$ has the form $\tfrac12 + i\gamma$ with $\gamma \in \mathbb{R}$; below $\gamma$ and $\gamma'$ always denote imaginary parts of non-trivial zeros of $\zeta$ (counted with multiplicity).

For fixed real numbers $\alpha \le \beta$ put $\delta(\alpha,\beta) = 1$ if $0 \in [\alpha,\beta]$ and $\delta(\alpha,\beta) = 0$ otherwise. Then

$$\lim_{T\to\infty} \frac{\#\left\{(\gamma,\gamma') : 0 < \gamma, \gamma' \le T \text{ and } \dfrac{2\pi\alpha}{\log T} \le \gamma - \gamma' \le \dfrac{2\pi\beta}{\log T}\right\}}{\dfrac{T}{2\pi}\log T} = \int_{\alpha}^{\beta}\left(1 - \left(\frac{\sin(\pi u)}{\pi u}\right)^{2}\right)\mathrm{d}u + \delta(\alpha,\beta),$$

where the integrand is interpreted by continuity at $u = 0$.

Equivalently, in terms of the Fourier-transform side: for $T > 2$ and $\alpha \in \mathbb{R}$, with the weight $w(u) := \dfrac{4}{4+u^{2}}$, set

$$F(\alpha) := F_{T}(\alpha) = \left(\frac{T}{2\pi}\log T\right)^{-1} \sum_{0 < \gamma, \gamma' \le T} T^{\,i\alpha(\gamma-\gamma')}\, w(\gamma-\gamma').$$

Then for every $A > 1$,

$$F_{T}(\alpha) = 1 + o(1), \qquad T \to \infty,$$

uniformly for $1 \le |\alpha| \le A$.