On small distances between ordinates of zeros of $\zeta(s)$ and   $\zeta'(s)$
  We prove that for any zero $\beta'+i\gamma'$ of $\zeta'(s)$ there exists a zero $\beta+i\gamma$ of $\zeta(s)$ such that $|\gamma-\gamma'|\ll \sqrt{|\beta'-\tfrac{1}{2}|},$ and we provide some other related results. 
M. Z. Garaev and C. Y. Yildirim