On the Zeros of ζ(s) and ζ′(s)

AbstractIn this paper, we will show that there is a close connection between the vertical distribution of the zeros of the Riemann Zeta function ζ(s) and that of the zeros of its derivative ζ′(s). To be precise, we will prove, assuming the Riemann Hypothesis, the following theorem: THEOREM. Let T0be a large fixed positive real number and T ≥ T0. Let ρ′0 = β′0 + iγ′0be a zero of ζ′(s) with 1/2 < β′0 < 1/2 + g(T) where g(T) → 0 when T → ∞, and T ≤ γ′0 ≤ 2T. Suppose there exists another zero ρ′1 of ζ′(s) with |ρ′1 − ρ′0| ≤ A(β′0 − 1/2) for some absolute constant A > 0. Then there exists a positive real number B depending only on A and a zero ρ = β + iγ of ζ(s) with |γ′0 − γ| ≤ B(β′0 − 1/2) (Note that here ρ′1may be equal to ρ′0, i.e., ζ′(s) may have a double zero at s = ρ′0.) We will also give a slight generalization of this result