The distribution of coefficients attached to the Dedekind zeta function over certain sparse sequences

Let \(K_{3}\) be a non-normal cubic extension over \(\mathbb{Q}\), and let \(a_{K_{3}}(n)\) be the \(n\)-th coefficient of the Dedekind zeta function \(\zeta_{K_{3}}(s)\). In this paper, we investigate the asymptotic behaviour of the type

$$ \notag \sum_{n\leq x}a_{K_{3}}^{2}(n^{\ell}),$$

where \(\ell\geq 2\) is any fixed integer. As an application, we also establish the asymptotic formulae of the variance of \(a_{K_{3}}^{2}(n^{\ell})\). Furthermore, we also consider the asymptotic relations for shifted convolution sums associated to \(a_{K_{3}}(n)\) with classical divisor function.