Extreme values of derivatives of zeta and $L$-functions

It is proved that as $T\to \mathrm{\infty }$, uniformly for all positive integers $\mathrm{\ell }⩽({\log }_{3}T)\mathrm{/}({\log }_{4}T)$, we have \begin{equation*} \max_{T\leqslant t\leqslant 2T}\left|\zeta^{(\ell)}\Big(1+it\Big)\right| \geqslant \big(\mathbf Y_{\ell}+ o\left(1\right)\big)\left(\log_2 T \right)^{\ell+1} ,, \end{equation*} where $\mathbf Y_{\ell} = \int_0^{\infty} u^{\ell} \rho (u) du$\mathrm{.}Here$\rho(u)$istheDickmanfunction\mathrm{.}Wehave$\mathbf Y_{\ell} > e^{\gamma}/(\ell + 1)$and$ \log, \mathbf Y_{\ell} = \left(1 + o\left(1\right) \right) \ell \log \ell$when$ \ell \to \infty $, which significantly improves previous results in [17, 40]. Similar results are established for Dirichlet $L$-functions. On the other hand, when assuming the Riemann Hypothesis and the Generalized Riemann Hypothesis, we establish upper bounds for $\mid {\zeta }^{(\mathrm{\ell })}(1+it)\mid$ and $\mid L^{(\mathrm{\ell })}(1,\chi )\mid$. Furthermore, when assuming the Granville-Soundararajan Conjecture is true, we establish the following asymptotic formulas $$\underset{\begin{array}{c}\chi \mathrm{\ne }{\chi }_0\\ \chi (\text{mod}q)\end{array}}{\max }\mid L^{(\mathrm{\ell })}(1,\chi )\mid \sim {\mathbf{Y}}_{\mathrm{\ell }}{\left({\log }_{2}q\right)}^{\mathrm{\ell }+1},\text{as}q\to \mathrm{\infty },$$ where $q$ is prime and $\mathrm{\ell }\in \mathbb{N}$ is given.