On the Number of Integral Ideals in Two Different Quadratic Number Fields

Let $K$ be an algebraic number field of finite degree over the rational field $\mathbb{Q}$, and $a_K(n)$ the number of integral ideals in $K$ with norm $n$. When $K$ is a Galois extension over $\mathbb{Q}$, many authors contribute to the integral power sums of $a_K(n)$, \begin{align*} \sum_{n \leq x}a_K(n)^l, \quad l=1,2,3,\cdots. \end{align*} 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 \begin{align*} \sum_{n\leq x}a_{K_{1}}(n^j)^la_{K_{2}}(n^j)^l, \quad j=1,2,l=2,3,\cdots, \end{align*} where $K_1$ and $K_2$ are two different quadratic fields.