Zeros of the derivatives of the Riemann zeta function on \(\operatorname{Re}s<1/2\) (Q2436761)

Let \(\zeta(s)\) denote the Riemann zeta-function, and let \(N_k^-(\sigma,T,H)\) denote the number of zeros \(\rho\) of \(\zeta^{(k)}(s)\) in the rectangle \(0 < \text{Re}(\rho) < \sigma\), \(T < \text{Im}(\rho) < T+H\). \textit{A. Speiser} [Math. Ann. 110, 514--521 (1934; Zbl 0010.16401, JFM 60.0272.04)] proved that the Riemann Hypothesis (that all the zeros of \(\zeta(s)\) lie in the half-plane Re\((s) \leq 1/2\)) is equivalent to the non-vanishing of \(\zeta'(s)\) in the strip \(0 < \text{Re}(s) < 1/2\). Motivated by this result, \textit{N. Levinson} and \textit{H. L. Montgomery} [Acta Math. 133, 49--65 (1974; Zbl 0287.10025)] proved that if \(w(T) \to \infty\) as \(T \to \infty\), \(\delta(T) = w(T)/\ln T\), and \(1/2 < a \leq 1\), then \[ N_1^-(1/2-\delta(T), T, T^a) \ll T^{a-(2a-1)\delta(T)/4}w(T)^2\ln T. \] In the paper under review, the author extends the case \(a=1\) of this bound to higher derivatives of the zeta-function. Let \(k > 1\) be an integer, and let \(A\) and \(c\) be reals with \(0 < c < 1/2\). The main result of the paper then states that if \(T \to \infty\) and \(\sigma \leq 1/2 - A(\ln\ln T)/\ln T\), then \[ N_k^-(\sigma, T, T) \ll T^{1 + c(\sigma-1/2)}\frac {(\ln T)^3\ln\ln\ln T}{\ln\ln T}. \]