Extreme values of derivatives of the Riemann zeta function

Abstract

It is proved that if $T$ is sufficiently large, then uniformly for all positive integers $\ell \leqslant (\log T) / (\log_2 T)$, we have \begin{equation*} \max_{T\leqslant t\leqslant 2T}\left|\zeta^{(\ell)}\Big(1+it\Big)\right| \geqslant e^{\gamma}\cdot \ell^{\ell}\cdot (\ell+1)^{ -(\ell+1)}\cdot\Big(\log_2 T - \log_3 T + O(1)\Big)^{\ell+1} \,, \end{equation*} where $\gamma$ is the Euler constant. We also establish lower bounds for maximum of $\big|\zeta^{(\ell)}(\sigma+it)\big|$ when $\ell \in \mathbb N $ and $\sigma \in [1/2, \,1)$ are fixed.