Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Ekta Soni, M. S. Datt, A. Sankaranarayanan

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.

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

Abstract

Let be a non-normal algebraic number field of cubic degree given by the polynomial of discriminant . For sufficiently large , we establish an asymptotic formula for the hybrid sum with a tight error term.
Paper Structure (3 sections, 8 theorems, 42 equations)

This paper contains 3 sections, 8 theorems, 42 equations.

Table of Contents

  1. Introduction
  2. Preliminaries and Some Lemmas
  3. Proof of the main Theorem

Key Result

Lemma 2.1

Let $g(n)=(-1)^{n}\sum_{d|n}(-1)^{d}d^{3}$. Then $g(n)$ is a multiplicative arithmetic function.

Theorems & Definitions (10)

  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • Lemma 2.4
  • Lemma 2.5
  • Lemma 2.6
  • Lemma 2.7
  • Lemma 2.8