On the distribution of zeros of derivatives of the Riemann ζ-function

For the completed Riemann zeta function ζ (s) , it is known that the Riemann hypothesis for ζ (s) implies the Riemann hypothesis for ζ (m) (s) , where m is any positive integer. In this paper, we investigate the distribution of the fractional parts of the sequence (α γ m) (αγm), where α is any fixed non-zero real number and γ m γm runs over the imaginary parts of the zeros of ζ (m) (s) . We also obtain a zero density estimate and an explicit formula for the zeros of ζ (m) (s). In particular, all our results hold uniformly for 0 ≤ m ≤ g (T) 0 ≤ q m , where the function g (T) tends to infinity with T and g (T) = o (T) .