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

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 $\mathrm{\ell }$ be a fixed natural number. We show that, if $\mathrm{\mid }\sigma -1\mathrm{\mid }\ll 1\mathrm{/}{\log }_{2}t$, then $\mathrm{\mid }{\zeta }^{(\mathrm{\ell })}(\sigma +it)\mathrm{\mid }$ has the same maximal order (up to the leading coefficients) as $\mathrm{\mid }{\zeta }^{(\mathrm{\ell })}(1+it)\mathrm{\mid }$ when $t\to \mathrm{\infty }$. The range $1-\sigma \ll 1\mathrm{/}{\log }_{2}t$ is wide enough, since we also show that $(1-\sigma ){\log }_{2}t\to \mathrm{\infty }(t\to \mathrm{\infty })$ implies ${\mathrm{lim\hspace{0.17em}sup}}_{t\to \mathrm{\infty }}\mathrm{\mid }{\zeta }^{(\mathrm{\ell })}(\sigma +it)\mathrm{\mid }\mathrm{/}({\log }_{2}t{)}^{\mathrm{\ell }+1}=\mathrm{\infty }$. Similar results can be obtained for Dirichlet $L$-functions $L^{(\mathrm{\ell })}(\sigma ,\chi )$ with $\chi (\mathrm{m}\mathrm{o}\mathrm{d}q)$.