Let \(K_{3}\) be a non-normal cubic extension over \(\mathbb {Q}\) , and let \(\tau _{k}^{K_{3}}(n)\) denote the k-dimensional divisor function in the number field \(K_{3}/\mathbb {Q}\) . In this paper, we investigate the asymptotic behaviour of the general divisor problem involving several different forms of \((\tau _{k}^{K_{3}}(n))^{\ell }\) with \(k,\ell \geqslant 2\) . As an application, we also obtain the asymptotic formulae of the variance of these coefficients associated to the Dedekind zeta function of \(K_{3}/\mathbb {Q}\) . By analogy, we also consider the analogous problem supported on the set of certain integral binary quadratic forms.
