Let \(K_{3}/\mathbb {Q}\) be a non-normal cubic extension, and let \(\tau _{k}^{K_{3}}(n)\) denote the k-dimensional divisor function in the number field \(K_{3}/\mathbb {Q}\) , for any fixed integer \(k\ge 1\) . In this paper, we investigate the asymptotic behaviour of a general divisor function involving several different forms of \((\tau _{k}^{K_{3}}(n))^{\ell }\) with \(k\ge 2,\ell \ge 1\) , supported on the sequence of positive integers represented by primitive integral positive definite reduced binary quadratic forms with a fixed discriminant \(\mathfrak {D}<0\) . As an application, we also obtain the asymptotic formulae of the variance of these coefficients.
