To study the distribution of pairs of zeros of the Riemann zeta-function, Montgomery introduced the function$$ F(\alpha) = F_T(\alpha) = \left({T\over 2\pi}\log T\right)^{-1} \sum_{0<\gamma,\gamma ' \le T} T^{i\alpha(\gamma -\gamma ')}w(\gamma-\gamma '),$$where $\alpha$ is real and $T\ge 2$, $\gamma$ and $\gamma '$ denote the imaginary parts of zeros of the Riemann zeta-function,and $w(u) = 4/(4 + u^2)$.Assuming the Riemann Hypothesis, Montgomery proved an asymptotic formula for $F(\alpha)$ when $|\alpha|\le 1$, and made the conjecturethat $F(\alpha) = 1 + o(1)$ as $T\to \infty$for any bounded $\alpha$ with $|\alpha |\ge 1$.In this paper we use an approximation for the primeindicator function together with a new mean valuetheorem for long Dirichlet polynomials and tails ofDirichlet series to prove that, assuming the GeneralizedRiemann Hypothesis for all Dirichlet $L$-functions, thenfor any $\epsilon >0$ we have$$ F(\alpha) \ge {3\over 2} -|\alpha| - \epsilon ,$$ uniformly for $1\le |\alpha| \le\frac32 -2\epsilon $and all $T \ge T_0(\epsilon)$.

1991 Mathematics Subject Classification: primary 11M26; secondary 11P32.