  We make explicit an argument of Heath-Brown concerning large and small gaps
between nontrivial zeroes of the Riemann zeta-function, $\zeta(s)$. In
particular, we provide the first unconditional results on gaps (large and
small) which hold for a positive proportion of zeroes. To do this we prove
explicit bounds on the second and fourth power moments of $S(t+h)-S(t)$, where
$S(t)$ denotes the argument of $\zeta(s)$ on the critical line and $h \ll 1 /
\log T$. We also use these moments to prove explicit results on the density of
the nontrivial zeroes of $\zeta(s)$ of a given multiplicity.
