On the number of integral ideals in two different quadratic number fields

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 / Integral ideal / Number field

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