Inequalities of Ono numbers and class numbers associated to imaginary quadratic fields

Abstract

We denote by $h_{D}$ the class number and by $p_{D}$ the Ono number of the imaginary quadratic fields $\mathbf{Q}(\sqrt{-D})$. Sairaiji-Shimizu [2] showed that there are infinitely many imaginary quadratic fields such that the inequality $h_{D} >c^{ p_{D}}$ holds for any real number. On the other hand we have the possibility that $h_{D} \leqq c^{ p_{D}}$ holds for infinitely many imaginary quadratic fields for the same real number $c$. In this paper, given a real number $c$, we consider whether $h_{D} \leqq c^{ p_{D}}$ holds for infinitely many imaginary quadratic fields or not.