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

Maxim A. Korolev¹

¹ Steklov Mathematical Institute Russian Academy of Sciences Gubkin St., 8 119991 Moscow, Russia and National Research Nuclear University MEPhI (Moscow Engineering Physics Institute) Kashirskoye sh., 31 115409 Moscow, Russia

Acta Arithmetica, Tome 166 (2014) no. 4, pp. 349-390.

Résumé

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

Détail

DOI : 10.4064/aa166-4-3

Keywords: obtain series conditional lower bounds modulus argument riemann zeta function short segments critical line based riemann hypothesis particular prove large fixed constant there exist non effective constants maximum zeta interval t h greater