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, 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.

A limitation on proving the existence of small gaps between zeta-zeros

“…Trivially, we have that µ ≤ 1 ≤ λ, and it is expected that µ = 0 and λ = ∞. We refer the reader to [4,8] for the history of this problem. The best current results under RH are: µ ≤ 0.515396 by Preobrazhenski ȋ [7] and λ ≥ 3.18 by Bui and Milinovich [1].…”