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

Abstract

It is proved that as $T\to \infty$, uniformly for all positive integers $\ell ⩽({\log }_{3}T)/({\log }_{4}T)$, we have \begin{equation*} \max_{T\leqslant t\leqslant 2T}\left|ζ^{(\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 $$\underset{\begin{array}{c}\chi \ne {\chi }_0\\ \chi (\text{mod}q)\end{array}}{\max }\left|L^{(\ell )}(1,\chi )\right|\sim {\mathbf{Y}}_{\ell }{\left({\log }_{2}q\right)}^{\ell +1},\text{as}q\to \infty ,$$ where $q$ is prime and $\ell \in \mathbb{N}$ is given.

Version 4: 19 pages, to appear in BLMS. Dedicated to Professor K. Seip on the occasion of his 60th birthday

References in corpus (3)

• Large GCD sums and extreme values of the Riemann zeta function
• Extreme values of the Riemann zeta function and its argument
• Bounding $\zeta (s)$ in the critical strip