The zeros of the derivative of the Riemann zeta function near the critical line

Abstract

We study the horizontal distribution of zeros of $ζ&#39;(s)$ which are denoted as $ρ&#39;=β&#39;+iγ&#39;$. We assume the Riemann hypothesis which implies $β&#39;\geqslant1/2$ for any non-real zero $ρ&#39;$, equality being possible only at a multiple zero of $ζ(s)$. In this paper we prove that $\liminf(β&#39;-1/2)\logγ&#39;\not=0$ if and only if for any $c>0$ and $s=σ+it$ with $|σ-1/2|<c/\log t$ $(t\geqslant10)$ $$ \frac{ζ&#39;}ζ(s)=\frac{1}{s-ρ}+O(\log t), $$ where $ρ=1/2+iγ$ is the closest zero of $ζ(s)$ to $s$ and the origin. We also show that if $\liminf(β&#39;-1/2)\logγ&#39;\not=0$, then for any $c>0$ and $s=σ+it$ ($t\geqslant10$), we have $$ \logζ(s)=O(\frac{(\log t)^{2-2σ}}{\log\log t}) $$ uniformly for $1/2+c/\log t\leqslantσ\leqslantσ_1<1$.