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Divisibility of class numbers of imaginary quadratic fields. (English) Zbl 1018.11054

For any rational integer g2, let 𝒩 g (X) be the number of squarefree (positive) integer dX such that the ideal class group of the imaginary quadratic number field (-d) contains an element of order g. It is believed that 𝒩 g (X)C g X for some positive constant C g , however the asymptotic formula for 𝒩 g (X) is still unknown except for the case g=2, in which case we easily see 𝒩 2 (X)(6/π 2 )X by genus theory. The author improves the best known result 𝒩 g (X)X 1/2+1/g for general g3 due to M. Ram Murty [Topics in number theory, Kluwer Math. Appl., Dordr. 467, 229–239 (1999; Zbl 0993.11059)] to

𝒩 g (X)X 1/2+2/g-ε ifg0(mod4)


𝒩 g (X)X 1/2+3/(g+2)-ε ifg2(mod4)·

(Note that for odd g, we have 𝒩 g (X)𝒩 2g (X)X 1/2+3/(2(g+1))-ε .) He also offers a simple proof of 𝒩 4 (X)X/logX.

11R29Class numbers, class groups, discriminants
11R11Quadratic extensions