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

TL;DR

This paper derives an asymptotic formula with a tight error term for a hybrid sum involving the squared coefficients of a cubic non-normal algebraic number field, extending understanding of mean square behavior in algebraic number theory.

Contribution

It establishes a new asymptotic formula with a precise error term for a hybrid sum involving $a_{K}(n)$ in cubic non-normal algebraic number fields.

Findings

Asymptotic formula for the hybrid sum is proven.

Error term in the asymptotic formula is tightly estimated.

Results extend mean square analysis to non-normal cubic fields.

Abstract

Let $\mathbb{K}$ be a non-normal algebraic number field of cubic degree given by the polynomial $x^{3}+ax^{2}+bx+c$ of discriminant $D_{\mathbb{K}}<0$. For sufficiently large $x$, we establish an asymptotic formula for the hybrid sum $$\sum\limits_{\substack{n= \sum_{i=1}^{8}n_{i}^{2}\leq x\\ (n_{1},n_{2},n_{3},n_{4},n_{5},n_{6},n_{7},n_{8})\in \mathbb{Z}^{8} }} a_{\mathbb{K}}^{2}(n)$$ with a tight error term.