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

2020/10/20 by Simonič, A., Trudgian, T., Turnage-Butterbaugh, C. L.
#11M06 #11M26 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2010.10675

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.

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