  In this paper we study the distribution of the non-trivial zeros of the
Riemann zeta-function $\zeta(s)$ (and other L-functions) using Montgomery's
pair correlation approach. We use semidefinite programming to improve upon
numerous asymptotic bounds in the theory of $\zeta(s)$, including the
proportion of distinct zeros, counts of small gaps between zeros, and sums
involving multiplicities of zeros.
