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

Abstract

We study the horizontal distribution of zeros of ${\zeta }^{\prime }(s)$ which are denoted as ${\rho }^{\prime }={\beta }^{\prime }+i{\gamma }^{\prime }$. We assume the Riemann hypothesis which implies ${\beta }^{\prime }⩾1/2$ for any non-real zero ${\rho }^{\prime }$, equality being possible only at a multiple zero of $\zeta (s)$. In this paper we prove that $\mathrm{liminf}({\beta }^{\prime }-1/2)\log {\gamma }^{\prime }\ne 0$ if and only if for any $c>0$ and $s=\sigma +it$ with $|\sigma -1/2|<c/\log t$ $(t⩾10)$ $$\frac{{\zeta }^{\prime }}{\zeta }(s)=\frac{1}{s-\rho }+O(\log t),$$ where $\rho =1/2+i\gamma$ is the closest zero of $\zeta (s)$ to $s$ and the origin. We also show that if $\mathrm{liminf}({\beta }^{\prime }-1/2)\log {\gamma }^{\prime }\ne 0$, then for any $c>0$ and $s=\sigma +it$ ($t⩾10$), we have $$\log \zeta (s)=O(\frac{(\log t{)}^{2-2\sigma }}{\log \log t})$$ uniformly for $1/2+c/\log t⩽\sigma ⩽{\sigma }_1<1$.

19 pages