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

Abstract

It is proved that as $T \to \infty$, uniformly for all positive integers $\ell \leqslant (\log_3 T) / (\log_4 T)$, 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 $\mathbf Y_{\ell} = \int_0^{\infty} u^{\ell} \rho (u) du$. Here $\rho(u)$ is the Dickman function. We have $\mathbf Y_{\ell} > e^{\gamma}/(\ell + 1)$ 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 $ \left| \zeta^{(\ell)}\left(1+it\right)\right| $ and $\left|L^{(\ell)}(1, \chi) \right|$. Furthermore, when assuming the Granville-Soundararajan Conjecture is true, we establish the following asymptotic formulas $$\max_{ \substack{ \chi \neq \chi_0 \\ \chi(\text{mod}\, q)}} \left|L^{(\ell)}(1, \chi) \right| \sim \mathbf Y_{\ell}\left(\log_2 q\right)^{\ell+1},\,\, \quad \text{as}\,\quad q \to \infty,$$ where $q$ is prime and $\ell \in \mathbb{N}$ is given.