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

Volume 166 / 2014

Maxim A. Korolev Acta Arithmetica 166 (2014), 349-390 MSC: Primary 11M06. DOI: 10.4064/aa166-4-3


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 $h = (1/\pi)\ln\ln\ln{T}+c_{0}$.


  • Maxim A. KorolevSteklov Mathematical Institute
    Russian Academy of Sciences
    Gubkin St., 8
    119991 Moscow, Russia
    National Research Nuclear University
    (Moscow Engineering Physics Institute)
    Kashirskoye sh., 31
    115409 Moscow, Russia

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