Conditional estimates on small distances between ordinates of zeros of ζ(s) and ζ′(s)

Let β′+iγ′β′+iγ′ be a zero of ζ′(s)ζ′(s). In [3] Garaev and Yıldırım proved that there is a zero β+iγβ+iγ of ζ(s)ζ(s) with γ′−γ≪|β′−1/2|. Assuming RH, we improve this bound by saving a factor log⁡log⁡γ′.