An inequality between class numbers and Ono's numbers associated to imaginary quadratic fields. (Q1869646)

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.