Omega-theorems for Riemann's zeta function and its derivatives near the line $\operatorname{Re}s=1$

Theorem of Zaitsev [1] states that
$$ \limsup_{s \in \Sigma(T),\;T\to +\infty}\frac{|\zeta(s)|}{\ln T}\geqslant1, $$
where $\Sigma(T)$ denotes the domain
$$ \quad 1-(4+\varepsilon)\frac{\ln\ln\ln t}{\ln\ln t}\leqslant \sigma \leqslant 1,\quad t_{0}<|t|\leqslant T. $$
In this talk, we will present a generalization of Zaitsev's method that allows us to obtain a family of omega-theorems for the Riemann's zeta function and its derivatives in various domains of the critical strip. In particular, we were able to prove that in the same domain $\Sigma(T)$ for all $n$ and arbitrary positive $\delta$ the inequality
$$ \limsup_{s \in \Sigma(T),\;T\to +\infty} \frac{|\zeta^{(n)}(s)|}{e^{(\ln\ln T)^{1+\varepsilon/2-\delta}}}\geqslant1, $$
holds.