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.
Citation: Qian Wang, Xue Han. Comparing the number of ideals in quadratic number fields[J]. Mathematical Modelling and Control, 2022, 2(4): 268-271. doi: 10.3934/mmc.2022025
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.
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Qian Wang, Xue Han. Comparing the number of ideals in quadratic number fields[J]. Mathematical Modelling and Control, 2022, 2(4): 268-271. doi: 10.3934/mmc.2022025
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