  We study three integrals related to the celebrated pair correlation
conjecture of H. L. Montgomery. The first is the integral of Montgomery's
function $F(\alpha, T)$ in bounded intervals, the second is an integral
introduced by Selberg related to estimating the variance of primes in short
intervals, and the last is the second moment of the logarithmic derivative of
the Riemann zeta-function near the critical line. The conjectured asymptotic
for any of these three integrals is equivalent to Montgomery's pair correlation
conjecture. Assuming the Riemann hypothesis, we substantially improve the known
upper and lower bounds for these integrals by introducing new connections to
certain extremal problems in Fourier analysis. In an appendix, we study the
intriguing problem of establishing the sharp form of an embedding between two
Hilbert spaces of entire functions naturally connected to Montgomery's pair
correlation conjecture.
