On the distribution of values of the derivative of the Riemann zeta function at its zeros. I

Let ζ′(s) be the derivative of the Riemann zeta function ζ(s). A study on the value distribution of ζ′(s) at the non-trivial zeros ρ of ζ(s) is presented. In particular, for a fixed positive number X, an asymptotic formula and a non-trivial upper bound for the sum Σ0<Im ρ≤Tζ′(ρ)Xρ as T → ∞ are given. We clarify the dependence on the arithmetic nature of X.