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 $|\sigma -1|\ll 1/{\log }_{2}t$, 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 $1-\sigma \ll 1/{\log }_{2}t$ is wide enough, since we also show that $(1-\sigma ){\log }_{2}t\to \infty (t\to \infty )$ implies ${\mathrm{limsup}}_{t\to \infty }|{\zeta }^{(\ell )}(\sigma +it)|/({\log }_{2}t{)}^{\ell +1}=\infty$. Similar results can be obtained for Dirichlet $L$-functions $L^{(\ell )}(\sigma ,\chi )$ with $\chi (\mathrm{mod}q)$.

arXiv comment18 pages