The general divisor problem of higher moments of coefficients attached to the Dedekind zeta function

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 higher moments of the coefficients attached to the Dedekind zeta function over sum of two squares of the form

$$\begin{aligned} \sum _{n_{1}^{2}+n_{2}^{2}\le x}(\tau _{k}^{K_{3}}(n_{1}^{2}+n_{2}^{2}))^{l}, \end{aligned}$$

where \(n_{1}, n_{2}\in \mathbb {Z}\), and \(k\ge 2, l\ge 2\) are any fixed integers.