Extreme values of derivatives of the Riemann zeta function

TL;DR

This paper establishes new lower bounds for the maximum magnitude of derivatives of the Riemann zeta function along the critical line and in the half-plane, revealing extreme value behavior for large T and derivatives order.

Contribution

It provides the first uniform lower bounds for derivatives of the zeta function on the critical line and extends results to fixed real parts between 1/2 and 1, advancing understanding of their extremal values.

Findings

Proved lower bounds for derivatives of zeta on the critical line for large T.

Established bounds for derivatives with fixed real part in [1/2, 1).

Quantified the growth of zeta derivatives in terms of T and derivative order.

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.