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Divisibility of class numbers of imaginary quadratic fields. (English) Zbl 1018.11054
For any rational integer \(g\geq 2\), let \(\mathcal{N}_{g}(X)\) be the number of squarefree (positive) integer \(d\leq X\) such that the ideal class group of the imaginary quadratic number field \({\mathbb Q}(\sqrt{-d})\) contains an element of order \(g\). It is believed that \(\mathcal{N}_{g}(X)\sim C_{g} X\) for some positive constant \(C_{g}\), however the asymptotic formula for \(\mathcal{N}_{g}(X)\) is still unknown except for the case \(g=2\), in which case we easily see \(\mathcal{N}_{2}(X)\sim (6/\pi^{2})X\) by genus theory. The author improves the best known result \(\mathcal{N}_{g}(X)\gg X^{1/2+1/g}\) for general \(g\geq 3\) due to M. Ram Murty [Topics in number theory, Kluwer Math. Appl., Dordr. 467, 229–239 (1999; Zbl 0993.11059)] to \[ \mathcal{N}_{g}(X)\gg X^{1/2+2/g-\varepsilon}\quad\text{if}\;g\equiv 0\pmod{4} \] and \[ \mathcal{N}_{g}(X)\gg X^{1/2+3/(g+2)-\varepsilon}\quad\text{if}\;g\equiv 2\pmod{4}. \] (Note that for odd \(g\), we have \(\mathcal{N}_{g}(X)\geq \mathcal{N}_{2g}(X)\gg X^{1/2+3/(2(g+1))-\varepsilon}\).) He also offers a simple proof of \(\mathcal{N}_{4}(X)\gg X/\sqrt{\log X}\).

MSC:
11R29 Class numbers, class groups, discriminants
11R11 Quadratic extensions
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