Imaginary quadratic fields with class groups of 3-rank at least 2
Let \(N_k(x)\) denote the number of imaginary quadratic fields \(\mathbb Q(\sqrt{-d})\) with \(d\le x\) and the \(3\)-rank of the class-group being \(\ge k\). In [Ramanujan J. 11, No. 2, 159--163 (2006; Zbl 1161.11391)] \textit{D. Byeon} established \[ N_2(x)\gg x^{1/3-\varepsilon} \] for every \(\varepsilon\), and later \textit{F. Luca} and \textit{A. M. Pacelli} [J. Number Theory 128, No. 4, 796--804 (2008; Zbl 1167.11038)] removed the \(\varepsilon\) from the exponent. Now the author improves this bound to \[ N_2(x)\gg x^{1/2-\varepsilon} \] for all \(\varepsilon>0\), utilizing a theorem of \textit{C. Hooley} [in: Analytic number theory. Essays in honour of Klaus Roth on the occasion of his 80th birthday. Cambridge: Cambridge University Press. 235--266 (2009; Zbl 1226.11098)] on square-free values of polynomials in two variables. Similar bounds for larger values of \(k\) have been obtained recently by \textit{A. Levin} et al. [Acta Arith. 197, 275--292 (2021)] who showed \[ N_3(x)\gg \frac{x^{1/9}}{\log x} \] and \[ N_5(x)\gg \frac{x^{1/30}}{\log x}. \]
scientific article; zbMATH DE number 7260774
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