Short series over simple zeros of the Riemann zeta-function.
It was shown by \textit{M. Z. Garaev} [Math. Notes 73, 585--587 (2003; Zbl 1101.11030)] that \[ \sum_{0< \gamma\leq T}|\zeta'(\rho)|^{-1}\gg(\log T)^{1/2}, \] where the sum runs over simple zeros \(\rho= \beta+ i\gamma\) of the Riemann zeta-function. In the present paper the lower bound is improved to \((\log T)^{3/4}\). Indeed the corresponding sum over a short interval is investigated, and it is shown that \[ \sum_{T<\gamma\leq T+ H}|\zeta'(\rho)|^{-1}\gg H(\log T)^{-1/4}, \] for \(T^{0.552}\leq H\leq T\), from which the stated result follows. The key tools for the proof are the facts that \[ N_1(T+ H)- N_1(T)\gg H\log T \] for \(T^{0.552}\leq H\leq T\), where \(N_1(T)\) counts simple zeros on the critical line [see \textit{J. Steuding}, Acta Math. Hungar. 96, No. 4, 259--308 (2002; Zbl 1012.11080)] and that \[ \int^{T+H}_T |\zeta''(1/2+ it)|\,dt\ll H(\log T)^{9/4} \] for \(H\geq T^{1/2}+\varepsilon\) [see \textit{K. Ramachandra}, Ann. Acad. Sci. Fenn., Ser. A I 5, 145--158 (1980; Zbl 0411.10015 and Zbl 0448.10031)].
scientific article; zbMATH DE number 2124634