Extreme values of the Dedekind zeta function on the critical line

TL;DR

This paper improves bounds on the maximum size of Dedekind zeta functions on the critical line for number fields, using asymptotic sum estimates related to Gál sums, extending previous results.

Contribution

It provides a new lower bound for the large values of Dedekind zeta functions on the critical line, refining earlier bounds by employing advanced asymptotic sum analysis.

Findings

Established a new lower bound for Dedekind zeta function maxima

Extended the range of the degree d for which bounds hold

Improved understanding of zeta function behavior on the critical line

Abstract

By employing the assessment of the asymptotic size of various sums of G\'{a}l studied by La Bret\`eche and Tenenbaum, we provide an improvement on the recent result of A. Bondarenko, P. Darbar, M. V. Hagen, W. Heap, and K. Seip regarding the large values of the Dedekind zeta-function on the critical line. Specifically, let $d\geqslant 3$ be an integer and $A$ be a positive constant. Denoting $K=\mathbb{Q}(\zeta_d)$, we establish that, if $T$ is sufficiently large, then uniformly for $d \ll (\log\log T)^A$, \begin{equation*} \max_{ t \in [0,T]}\left|\zeta_K \left(\frac{1}{2}+it \right) \right| \gg \exp\left({(1+o(1))\varphi(d)} \sqrt{\frac{\log T \log \log \log T}{\log \log T}} \right). \end{equation*}