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

Let \(K_{3}\) $K_3$ be a non-normal cubic extension over \(\mathbb{Q}\) $\mathbb{Q}$ , and let \(a_{K_{3}}(n)\) $a_{K_3}(n)$ be the \(n\) $n$ -th coefficient of the Dedekind zeta function \(\zeta_{K_{3}}(s)\) ${\zeta }_{K_3}(s)$ . In this paper, we investigate the asymptotic behaviour of the type \( \notag \sum_{n\leq x}a_{K_{3}}^{2}(n^{\ell}),\) $$\sum _{n\le x}{a}_{K_3}^2(n^{\ell }),$$ where \(\ell\geq 2\) $\ell \ge 2$ is any fixed integer. As an application, we also establish the asymptotic formulae of the variance of \(a_{K_{3}}^{2}(n^{\ell})\) $a_{K_3}^2(n^{\ell })$ . Furthermore, we also consider the asymptotic relations for shifted convolution sums associated to \(a_{K_{3}}(n)\) $a_{K_3}(n)$ with classical divisor function.