Extreme values of the Dedekind zeta function on the critical line

Abstract

By employing the assessment of the asymptotic size of various sums of Gál studied by La Bretèche 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 $d⩾3$ 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|ζ_K \left(\frac{1}{2}+it \right) \right| \gg \exp\left({(1+o(1))φ(d)} \sqrt{\frac{\log T \log \log \log T}{\log \log T}} \right). \end{equation*}

12 pages