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 aK(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 aK(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 K1 and K2 are two different quadratic fields.