The distribution of zeros of ζ′(s) and gaps between zeros of ζ(s)

Abstract

Assume the Riemann Hypothesis. We establish a local structure theorem for zeros of the Riemann zeta-function ζ(s) and its derivative ζ′(s). As an application, we prove a stronger form of half of a conjecture of Radziwiłł [18] concerning the global statistics of these zeros. Roughly speaking, we show that on the Riemann Hypothesis, if there occurs a small gap between consecutive zeta zeros, then there is exactly one zero of ζ′(s) lying not only very close to the critical line but also between that pair of zeta zeros. This refines a result of Zhang [22]. Some related results are also shown. For example, we prove a weak form of a conjecture of Soundararajan, and suggest a repulsion phenomena for zeros of ζ′(s).