On the discrete mean square of certain hybrid sum involving $a_{\mathbb{K}}(n)$

Abstract

Let $\mathbb{K}$ be a non-normal algebraic number field of cubic degree given by the polynomial $x^3+a{x}^2+bx+c$ of discriminant $D_{\mathbb{K}}<0$. For sufficiently large $x$, we establish an asymptotic formula for the hybrid sum $$\sum _{\begin{array}{c}n={\sum }_{i=1}^8{n}_i^2\le x\\ (n_1,n_2,n_3,n_4,n_5,n_6,n_7,n_8)\in {\mathbb{Z}}^8\end{array}}{a}_{\mathbb{K}}^2(n)$$ with a tight error term.