  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 $1/2+c/\log
t\leqslant\sigma\leqslant\sigma_1<1$.
