  Montgomery's pair correlation conjecture predicts the asymptotic behavior of
the function $N(T,\beta)$ defined to be the number of pairs $\gamma$ and
$\gamma'$ of ordinates of nontrivial zeros of the Riemann zeta-function
satisfying $0<\gamma,\gamma'\leq T$ and $0 < \gamma'-\gamma \leq 2\pi
\beta/\log T$ as $T\to \infty$. In this paper, assuming the Riemann hypothesis,
we prove upper and lower bounds for $N(T,\beta)$, for all $\beta >0$, using
Montgomery's formula and some extremal functions of exponential type. These
functions are optimal in the sense that they majorize and minorize the
characteristic function of the interval $[-\beta, \beta]$ in a way to minimize
the $L^1\big(\mathbb{R}, \big\{1 - \big(\frac{\sin \pi x}{\pi x}\big)^2
\big\}\,dx\big)$-error. We give a complete solution for this extremal problem
using the framework of reproducing kernel Hilbert spaces of entire functions.
This extends previous work by P. X. Gallagher in 1985, where the case $\beta
\in \frac12 \mathbb{N}$ was considered using non-extremal majorants and
minorants.
