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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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