Comparing the number of ideals in quadratic number fields

Abstract

<abstract><p>Denote by $ a_{K}(n) $ the number of integral ideals in $ K $ with norm $ n $, where $ K $ is a algebraic number field of degree $ m $ over the rational field $ \mathcal{Q} $. Let $ p $ be a prime number. In this paper, we prove that, for two distinct quadratic number fields $ K_i = \mathcal{Q}(\sqrt{d_i}), \ i = 1, 2 $, the sets both</p> <p><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \{p\ |\ a_{K_1}(p)&lt; a_{K_2}(p)\} \text{ and } \{p\ |\ a_{K_1}(p^2)&lt; a_{K_2}(p^2)\} $\end{document} </tex-math></disp-formula></p> <p>have analytic density $ 1/4 $, respectively.</p></abstract>