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

Abstract

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.