Zeros of higher derivatives of Riemann zeta function (Q6614357)

In this article, the authors obtain an explicit constant in the error terms on zero density estimates of the \(k\)-th derivative of the Riemann zeta function \(\zeta^{(k)}(s)\) of \textit{N. Levinson} and \textit{H. Montgomery}'s results [Acta Math. 133, 49--65 (1974; Zbl 0287.10025), Theorem 5, Corollary of Theorem 4] as well as \textit{H. Ki} and \textit{Y. Lee}'s results [Funct. Approximatio, Comment. Math. 47, No. 1, 79--87 (2012; Zbl 1312.11068), Theorem 3 and Theorem 2]. Actually, the main concern of the paper is an upper bound for \[\sum_{\stackrel{T\leq \gamma_k\leq T+H}{\beta_k >\frac{1}{2}}} \left(\beta_k -\frac{1}{2}\right) \] and \[\sum_{\stackrel{T\leq \gamma_k\leq T+H}{\beta_k <\frac{1}{2}}} \left(\beta_k -\frac{1}{2}\right), \] where \(\rho_k=\beta_k+i\gamma_k\) is a zero of the \(k\)-th derivative of the Riemann zeta function \(\zeta^{(k)}(s)\).