Comparing the number of ideals in quadratic number fields

Abstract

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\ |\ a_{K_1}(p)< a_{K_2}(p)\} \text{ and } \{p\ |\ a_{K_1}(p^2)< a_{K_2}(p^2)\} $ have analytic density $ 1/4 $, respectively.

Keywords

• quadratic number fields
• prime
• dedekind zeta function