## An inequality between class numbers and Ono’s numbers associated to imaginary quadratic fields.(English)Zbl 1052.11070

Let $$k_ D$$ be an imaginary quadratic field with discriminant $$-D$$. We denote by $$h_ D$$ and by $$p_ D$$ the class number of $$k_ D$$ and Ono’s number of $$k_ D$$, respectively. In the previous paper [Proc. Japan Acad., Ser. A 77, No. 2, 29–31 (2001; Zbl 0988.11052)], the authors disproved Ono’s conjecture $$h_ D \leq 2^ {p_ D}$$ for all $$D$$ by showing that for any $$c>1$$ there exist infinitely many $$D$$ such that $$h_ D \geq c ^ {p_ D}$$. In this paper, they prove the modified inequality $$h_ D < {q_ D}^ {p_ D}$$ for all $$D$$, where $$q_ D$$ is the smallest prime number that splits completely in $$k_ D$$. (Note that $$q_ D=2$$ if and only if $$D\equiv 7\bmod 8)$$. They also discuss lower and upper bounds for $$p_ D$$ and give explicit bounds.

### MSC:

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

### Keywords:

Ono’s number; class number; imaginary quadratic field

Zbl 0988.11052
Full Text:

### References:

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