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Abstract
Let K be an algebraic number field of finite degree over the rational field Q, and a K(n) the number of integral ideals in K with norm n. When K is a Galois extension over Q, many authors contribute to the integral power sums of a K(n), $\sum\limits_{n \leqslant x} {a\kappa {{\left( n \right)}^l}} ,\;l = 1,\;2,\;3, \ldots $. This paper is interested in the distribution of integral ideals concerning different number fields. The author is able to establish asymptotic formulae for the convolution sum ${\sum\limits_{n \leqslant x} {a{\kappa _1}{{\left( {{n^j}} \right)}^l}a{\kappa _2}\left( {{n^j}} \right)} ^l},\;\;j = 1,\;2,\;\;l = \;2,\;3, \ldots $, where K 1 and K 2 are two different quadratic fields.
Keywords
Asymptotic formula
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Integral ideal
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Number field
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Zhishan Yang.
On the number of integral ideals in two different quadratic number fields.
Chinese Annals of Mathematics, Series B, 2016, 37(4): 595-606 DOI:10.1007/s11401-016-0963-7
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