Comparing the number of ideals in quadratic number fields

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.