On abelian cubic fields with large class number (Q6566377)

\textit{Y. Lamzouri} [Int. Math. Res. Not. 2015, No. 22, 11847--11860 (2015; Zbl 1376.11075)] proved that for a large real number \(x\), there are at least \(x^{1/2-1/{\log\log x}}\) real quadratic fields \(\mathbb{Q}(\sqrt{d})\) with discriminant \(d\leq x\) such that \N\[ h(d)\geq (2e^\gamma+o(1))\sqrt{d}\frac{\log\log d}{\log d}, \tag{1} \] \Nwhere \(h(d)\) and \(\gamma\) denote the class number of \(\mathbb{Q}(\sqrt{d})\) and the Euler-Mascheroni constant, respectively.\N\NIn the paper under review, the author extends $(1)$ to abelian cubic fields. The precise result is the following:\N\NLet \(\mathcal{F}(x)\) be the set of all simplest cubic field (in the sense of \textit{D. Shanks} [Math. Comput. 28, 1137--1152 (1974; Zbl 0307.12005)]). Then for a large real number \(x\), there are at least \(x^{1/4-o(1)}\) fields \(\mathbb{K}\) in \(\mathcal{F}(x)\) with discriminant \(d\leq x\) such that \[h_{\mathbb{K}}\geq \left(\frac{4}{91}e^{2\gamma}+o(1)\right)\sqrt{d}\left(\frac{\log\log d}{\log d}\right)^2,\] where \(h_\mathbb{K}\) is the class number of \(\mathbb{K}\).\N\NThe paper is well written and interesting to read.