Assume that \(K_3\) is a non-normal cubic extension over the rational field \(\mathbb {Q}\) . Denote by \(a_{K_3}(n)\) the number of integral ideals of the field \(K_3\) with norm n. Then \(a_{K_3}(n)\) can also be seen as the n-th Dirichlet coefficient of the Dedekind zeta function \(\zeta _{K_3}(s)\) . In this paper by introducing new ingredients, we first study the third moment of the Dirichlet coefficients \(a_{K_3}(n)\) and establish its asymptotic formula. Moreover, we also consider generalized divisor problems related to \(a_{K_3}(n)\) and establish their asymptotic formulae. Our results refine previous results.
