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А.A.Karatsuba's 80th Birthday Conference in Number Theory and Applications
May 22, 2017 12:15, Moscow, Steklov Mathematical Institute

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

A. B. Kalmynin

Department of Mathematics, National Research University "Higher School of Economics"
 Video records: MP4 550.7 Mb MP4 139.8 Mb

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Abstract: 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.

Language: English

References
1. S.P. Zaitsev, “Omega-teorema dlya dzeta-funktsii Rimana vblizi pryamoi $\operatorname{Re}s=1$”, Vestnik Moskovskogo un-ta. Ser. 1. Matematika. Mekhanika, 2000, № 3, 54–57; S.P. Zaitsev, “Omega-theorems for the Riemann zeta-function near the line $\operatorname{Re}s=1$”, Mosc. Univ. Math. Bulletin, 55:3 (2000)

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