Gaps between zeros of ζ(s)ζ(s) and the distribution of zeros of ζ′(s)ζ′(s)

We assume the Riemann Hypothesis in this paper. 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 ζ(s)ζ(s) if and only if there is a positive proportion of zeros of ζ′(s)ζ′(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 ζ′(s)ζ′(s). Finally on the Riemann Hypothesis and the Pair Correlation Conjecture we obtain near optimal bounds for the number of zeros of ζ′(s)ζ′(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.