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