Extreme values of the Riemann zeta function on the 1-line

By Christoph Aistleitner, Kamalakshya Mahatab, Marc Munsch. 2017

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We prove that there are arbitrarily large values of $t$ such that $|\zeta(1+it)| \geq e^{\gamma} (\log_2 t + \log_3 t) + \mathcal{O}(1)$. This essentially matches the prediction for the optimal lower bound in a conjecture of Granville and Soundararajan. Our proof uses a new variant of the "long resonator" method. While earlier implementations of this method crucially relied on a "sparsification" technique to control the mean-square of the resonator function, in the present paper we exploit certain self-similarity properties of a specially designed resonator function.

Subfields: Algebra and Number Theory, Discrete Mathematics and Combinatorics

View Paper: https://www.semanticscholar.org/paper/9f769f5d374401fb311bd4c94c55648138fc364e