Mean values related to the Dedekind zeta-function

summary:Let K/Q be a nonnormal cubic extension which is given by an irreducible polynomial g(x)=x³+a x²+b x+c. Denote by ζ K(s) the Dedekind zeta-function of the field K and aK(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 aK(n) is considered, i.e., ∑ n≤ xaK²(n)=x P₁(log x)+O(x5/7+ε), ∑ n≤ xaK³(n)=x P₄(log x)+O(X321/356+ε), where P₁(t), P₄(t) are polynomials of degree 1, 4, respectively, ε >0 is an arbitrarily small number.