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À.A.Karatsuba's 80th Birthday Conference in Number Theory and Applications
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Omega-theorems for Riemann's zeta function and its derivatives near the line A. B. Kalmynin Department of Mathematics, National Research University "Higher School of Economics" |
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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 $$ \quad 1-(4+\varepsilon)\frac{\ln\ln\ln t}{\ln\ln t}\leqslant \sigma \leqslant 1,\quad t_{0}<|t|\leqslant T. $$ 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 $$ \limsup_{s \in \Sigma(T),\;T\to +\infty} \frac{|\zeta^{(n)}(s)|}{e^{(\ln\ln T)^{1+\varepsilon/2-\delta}}}\geqslant1, $$ holds. Language: English References
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