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

11R29Class numbers, class groups, discriminants
11R11Quadratic extensions
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