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

TL;DR

This paper provides the first explicit, unconditional results on the distribution of gaps between nontrivial zeros of the Riemann zeta-function, including bounds on moments of argument differences and zero multiplicities.

Contribution

It establishes explicit bounds on moments of the argument of zeta and zero multiplicities, advancing understanding of zero gaps unconditionally.

Findings

Explicit bounds on second and fourth moments of $S(t+h)-S(t)$

Results on the density of zeros with given multiplicity

Unconditional results on large and small zero gaps

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.