On derivatives of zeta and $L$-functions near the 1-line

Zikang Dong, Yutong Song, Weijia Wang, Hao Zhang

Published 2023-12-19Version 1

We study the conditional upper bounds and extreme values of derivatives of the Riemann zeta function and Dirichlet $L$-functions near the 1-line. Let $\ell$ be a fixed natural number. We show that, if $|\sigma-1|\ll1/\log_2t$, then $|\zeta^{(\ell)}(\sigma+ i t)|$ has the same maximal order (up to the leading coefficients) as $|\zeta^{(\ell)}(1+ i t)|$ when $t\to\infty$. Similar results can be obtained for Dirichlet $L$-functions $L^{(\ell)}(\sigma,\chi)$ with $\chi\pmod q$.