Extreme values of the Dedekind zeta function 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*}