Montgomery's pair correlation conjecture for the nontrivial zeros of the Riemann zeta function

Let $\rho=\beta+i\gamma$ and $\rho'=\beta'+i\gamma'$ range over the nontrivial zeros of the Riemann zeta function. For fixed $\lambda>0$ and $T\to\infty$, consider pairs with positive ordinates at most $T$ and spacing $0<\gamma-\gamma'\leq 2\pi\lambda/\log T$. Montgomery's pair correlation conjecture.

$$\sum_{\substack{\rho,\rho' \\ 0<\gamma,\gamma'\le T \\ 0< \gamma-\gamma' \le \frac{2\pi\lambda}{\log T}}} 1 \sim \left(\int_0^\lambda\left(1-\left(\frac{\sin{\pi u}}{\pi u}\right)^2\right) du \right)\frac{T}{2\pi}\log T.$$

This is a conditional pair-correlation assertion used to study the distribution, simplicity, and location of zeta zeros. The source presents it as the needed asymptotic input for obtaining that asymptotically all zeros are simple and hence, by the preceding result, for addressing the proportion of zeros on the critical line; its resolution status is not specified here.