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

Abstract

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 $|\zeta^{(\ell)}(\sigma+ it)|$ has the same maximal order (up to the leading coefficients) as $|\zeta^{(\ell)}(1+ it)|$ when $t\to\infty$. The range $1-\sigma\ll1/\log_2t$ is wide enough, since we also show that $(1-\sigma) \log_2t \to \infty\; (t \to \infty)$ implies $\limsup_{t\to\infty}|\zeta^{(\ell)}(\sigma+ it)| / (\log_2t)^{\ell+1} = \infty$. Similar results can be obtained for Dirichlet $L$-functions $L^{(\ell)}(\sigma,\chi)$ with $\chi\pmod q$.