Gaps between zeros of $\zeta(s)$ and the distribution of zeros of $\zeta'(s)$

TL;DR

This paper establishes a connection between small gaps of zeros of the Riemann zeta function and zeros of its derivative, providing criteria related to the Siegel zero problem and bounds under the Riemann Hypothesis.

Contribution

It proves a conjecture linking zero gaps of z(s) and zeros of z'(s), with applications to the Siegel zero problem and bounds assuming RH and Pair Correlation.

Findings

Positive proportion of small gaps correlates with zeros of z'(s) near the half-line

Criterion for non-existence of Siegel zero based on zero distribution

Near optimal bounds for zeros of z'(s) under RH and Pair Correlation

Abstract

We settle a conjecture of Farmer and Ki in a stronger form. Roughly speaking we show that there is a positive proportion of small gaps between consecutive zeros of the zeta-function $\zeta(s)$ if and only if there is a positive proportion of zeros of $\zeta'(s)$ lying very closely to the half-line. Our work has applications to the Siegel zero problem. We provide a criterion for the non-existence of the Siegel zero, solely in terms of the distribution of the zeros of $\zeta(s)$. Finally on the Riemann Hypothesis and the Pair Correlation Conjecture we obtain near optimal bounds for the number of zeros of $\zeta'(s)$ lying very closely to the half-line. Such bounds are relevant to a deeper understanding of Levinson's method, allowing us to place one-third of the zeros of the Riemann zeta-function on the half-line.