Mean values related to the Dedekind zeta-function

Let K/ℚ be a nonnormal cubic extension which is given by an irreducible polynomial g(x) = x³ + ax² + bx + c. Denote by ζ_k(s) the Dedekind zeta-function of the field K and a_K(n) the number of integral ideals in K with norm n. In this note, by the higher integral mean values and subconvexity bound of automorphic L-functions, the second and third moment of a_K(n) is considered, i.e.

\(\sum_{n\leqslant x}a_K^2(n)=x P_1(\log x)+O(x^{5/7+\varepsilon}),\quad\sum_{n\leqslant x}a_K^3(n)=x P_4(\log x)+O(X^{321/356+\varepsilon}),\) $$\sum _{nx}{a}_K^2(n)=x{P}_1(\log x)+O(x^{5/7+\epsilon }),\sum _{nx}{a}_K^3(n)=x{P}_4(\log x)+O(X^{321/356+\epsilon }),$$

where P₁(t), P₄(t) are polynomials of degree 1, 4, respectively, ε > 0 is an arbitrarily small number.