Lower bounds for the maximum of the Riemann zeta function along vertical lines

TL;DR

This paper establishes new lower bounds for the maximum size of the Riemann zeta function along vertical lines in the critical strip, using a novel proof technique that also estimates the measure of large values.

Contribution

It introduces a different proof method for lower bounds of (+it), improving the understanding of its large values and their distribution.

Findings

Lower bounds for (+it) are established for large T.

The measure of t where |(+it)| is large is also bounded below.

The proof uses a modified resonance method and ideas from Hilberdink.

Abstract

Let $\alpha \in (1/2,1)$ be fixed. We prove that $$ \max_{0 \leq t \leq T} |\zeta(\alpha+it)| \geq \exp\left(\frac{c_\alpha (\log T)^{1-\alpha}}{(\log \log T)^\alpha}\right) $$ for all sufficiently large $T$, where we can choose $c_\alpha = 0.18 (2\alpha-1)^{1-\alpha}$. The same result has already been obtained by Montgomery, with a smaller value for $c_\alpha$. However, our proof, which uses a modified version of Soundararajan's "resonance method" together with ideas of Hilberdink, is completely different from Montgomery's. This new proof also allows us to obtain lower bounds for the measure of those $t \in [0,T]$ for which $|\zeta(\alpha+it)|$ is of the order mentioned above.