Some explicit and unconditional results on gaps between zeroes of the Riemann zeta-function

We make explicit an argument of Heath-Brown concerning large and small gaps between nontrivial zeroes of the Riemann zeta-function, ζ(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 ζ(s) on the critical line and h ≪ 1/ log T .We also use these moments to prove explicit results on the density of the nontrivial zeroes of ζ(s) of a given multiplicity.