Zeros of $\zeta^{\prime} (s)$ and the Riemann hypothesis

Abstract

It is shown that the Riemann hypothesis implies that the derivative of the Riemann zeta function has no zeros in the open left half of the critical strip. It is also shown, with no hypothesis, that, with the exception of a bounded region where the zeros can be calculated, the closed left half plane contains only real zeros of the derivative. It is further shown that the Riemann hypothesis is equivalent to the condition that $|\zeta(s)|$ increases as $\mathrm{Re}\,s$ moves left from $1/2$ for $\mathrm{Im}\,s$ sufficiently large.