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 Acta Arith., 2014, Volume 166, Issue 4, Pages 349–390 (Mi acta2)

On large values of the Riemann zeta-function on short segments of the critical line

M. A. Korolevab

a National Research Nuclear University MEPhI (Moscow Engineering Physics Institute), Kashirskoye sh., 31, 115409 Moscow, Russia
b Steklov Mathematical Institute, Russian Academy of Sciences, Gubkin St., 8, 119991 Moscow, Russia

Abstract: We obtain a series of new conditional lower bounds for the modulus and the argument of the Riemann zeta function on very short segments of the critical line, based on the Riemann hypothesis. In particular, we prove that for any large fixed constant $A>1$ there exist (non-effective) constants $T_0(A)>0$ and $c_0(A)>0$ such that the maximum of $|\zeta(0.5+it)|$ on the interval $(T-h,T+h)$ is greater than $A$ for any $T>T_0$ and $h=(1/\pi)\ln\ln\ln T+c_0$.

 Funding Agency Grant Number Russian Science Foundation 14-11-00433 The author is supported by Russian Scientific Fund (grant no. 14-11-00433).

DOI: https://doi.org/10.4064/aa166-4-3

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Document Type: Article
MSC: 11M06
Language: English