On derivatives of zeta and L-functions

Let \(\ell \) $\ell$ be a fixed natural number. We study the conditional upper bounds and extreme values of derivatives of the Riemann zeta function \(|\zeta ^{(\ell )}(\sigma +\textrm{i}t)|\) $|{\zeta }^{(\ell )}(\sigma +\text{i}t)|$ and Dirichlet L-functions \(L^{(\ell )}(\sigma ,\chi )\) $L^{(\ell )}(\sigma ,\chi )$ with \(\chi (\textrm{mod}\;q)\) $\chi (\text{mod}q)$ , where \(\sigma \) $\sigma$ is close to 1. We show that, if \(|\sigma -1|\ll 1/\log _2t\) $|\sigma -1|\ll 1/{log}_{2}t$ , then \(|\zeta ^{(\ell )}(\sigma +\textrm{i}t)|\) $|{\zeta }^{(\ell )}(\sigma +\text{i}t)|$ has the same maximal order (up to the leading coefficients) as \(|\zeta ^{(\ell )}(1+\textrm{i}t)|\) $|{\zeta }^{(\ell )}(1+\text{i}t)|$ when \(t\rightarrow \infty \) $t\to \infty$ . Similar results can be obtained for Dirichlet L-functions \(L^{(\ell )}(\sigma ,\chi )\) $L^{(\ell )}(\sigma ,\chi )$ with \(\chi \pmod q\) $\chi (modq)$ nonprincipal.