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

Abstract

We study the horizontal distribution of zeros of $\zeta'(s)$ which are denoted as $\rho'=\beta'+i\gamma'$. We assume the Riemann hypothesis which implies $\beta'\geqslant1/2$ for any non-real zero $\rho'$, equality being possible only at a multiple zero of $\zeta(s)$. In this paper we prove that $\liminf(\beta'-1/2)\log\gamma'\not=0$ if and only if for any $c>0$ and $s=\sigma+it$ with $|\sigma-1/2|<c/\log t$ $(t\geqslant10)$ $$ \frac{\zeta'}{\zeta}(s)=\frac{1}{s-\rho}+O(\log t), $$ where $\rho=1/2+i\gamma$ is the closest zero of $\zeta(s)$ to $s$ and the origin. We also show that if $\liminf(\beta'-1/2)\log\gamma'\not=0$, then for any $c>0$ and $s=\sigma+it$ ($t\geqslant10$), we have $$ \log\zeta(s)=O(\frac{(\log t)^{2-2\sigma}}{\log\log t}) $$ uniformly for $1/2+c/\log t\leqslant\sigma\leqslant\sigma_1<1$.