The zeros of the derivative of the riemann zeta function near the critical line

We study the horizontal distribution of zeros of ζ'(s) which are denoted as ρ' = β' + iγ'. We assume the Riemann hypothesis which implies β' ≥ 1/2 for any nonreal zero ρ', equality being possible only at a multiple zero of ζ(s). In this paper, we prove that lim inf (β' - 1/2) log γ' ≠ 0 if and only if, for any c> 0 and S = σ + it with 0 < |σ - 1/2| < c/ log t (t > t ₀(c)), we have ζ'/ζ(S)=1/S-ρ̂+O(log t), where ρ̂ = 1/2 + iγ is the zero of ζ closest to s (and to the origin, if there are two such). We also show that if lim inf [β' - 1/2) log γ' ≠0, then for any c> 0 and s = σ + it (t > t ₁(c)), we have logζ(S) = O ((log t)^2-2σ/log log t) uniformly for 1/2 + c/ log t ≤ σ < σ₁ 1.