On Titchmarsh's Phenomenon in the Theory of the Riemann Zeta Function

S. V. Konyagin ; M. A. Korolev

Informatics and Automation, Approximation Theory, Functional Analysis, and Applications, Tome 319 (2022), pp. 182-201

Résumé

We prove that the maximum modulus of the Riemann zeta function $\zeta (s)$ increases unboundedly when $s = 0.5+it$ varies on very short intervals of the critical line, and obtain an explicit lower bound for the growth rate of this maximum. This main result of the paper improves the second author's result of 2014 stating that this maximum becomes greater than any arbitrarily large fixed constant as $t$ increases. We also apply our method of proof to problems of large values of the argument of the zeta function and of irregularities in the distribution of the ordinates of zeros of $\zeta (s)$ on very short intervals of the critical line. We prove all these assertions assuming the Riemann hypothesis. The main ingredient of the method is an “effective” lemma on joint approximations of logarithms of prime numbers.

Détail

Keywords: Riemann zeta function, critical line, joint approximations, logarithms of primes
Mots-clés : Vinogradov cup.