On the zeros of $\zeta'(s)$ near the critical line

Abstract

Let $\rho^\prime=\beta^\prime+i\gamma^\prime$ denote the zeros of $\zeta^\prime(s),s=\sigma+it$. It is shown that there is a positive proportion of the zeros of $\zeta^\prime(s)$ in $0<t<T$ satisfying $\beta^\prime-1/2\ll(\log T)^{-1}$. Further results relying on the Riemann hypothesis and conjectures on the gaps between the zeros of $\zeta(s)$ are also obtained.