A note on the divisibility of class numbers of real quadratic fields.
\textit{Y. Yamamoto} [Osaka J. Math. 7, 57--76 (1970; Zbl 0222.12003)] and \textit{P. J. Weinberger} [J. Number Theory 5, 237--241 (1973; Zbl 0287.12007)] independently proved by different constructive methods that for any natural number \(g\geqq 3\) there exist infinitely many real quadratic fields whose class groups contain an element of order \(g\) [cf. \textit{H. Cohen} and \textit{H. W. Lenstra jun.}, Lect. Notes Math. 1068, 33--62 (1984; Zbl 0558.12002)]. It is expected that for the number \(S_{g}(X)\) of such fields with discriminant \(\leqq X\) we have \(S_{g}(X)\sim C_{g} X\) for some positive constant \(C_{g}\), however, the asymptotic formula for \(S_{g}(X)\) is still unknown even for a single \(g\geqq 3\). \textit{M. Ram Murty} [Topics in number theory, Kluwer Acad. Publishers. Math. Appl., Dordr. 467, 229--239 (1999; Zbl 0993.11059)] made a result of Weinberger effective to obtain \(S_{g}(X)\gg X^{1/2g-\varepsilon}\) for odd \(g\). The author improves this to \(S_{g}(X)\gg X^{1/g-\varepsilon}\) by estimating the number of discriminants based on a work of Y. Yamamoto. Similar results are known also for imaginary quadratic fields.
scientific article; zbMATH DE number 1891426
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