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On the Asymptotic Behavior of a Multiplicative Arithmetic Function Related to the Divisor Function Over Perfect Squares Integers Generated by Shifting

Bouderbala Mihoub

Abstract

Let $x$ be a real number satisfying $x \geq 2$. For any positive integer $n$, we define $s(n)$ as the smallest non-negative integer such that $n + s(n)$ is a perfect square. In this paper, we derive an asymptotic formula for the sum \begin{equation*} \sum_{n \leq x} D(n + s(n)), \end{equation*} where \begin{equation*} D(n) = \frac{τ(n)}{2^{ω(n)}}. \end{equation*} Here, $τ(n)$ denotes the number of positive divisors of $n$, and $ω(n)$ stands for the number of distinct prime factors of $n$.

On the Asymptotic Behavior of a Multiplicative Arithmetic Function Related to the Divisor Function Over Perfect Squares Integers Generated by Shifting

Abstract

Let be a real number satisfying . For any positive integer , we define as the smallest non-negative integer such that is a perfect square. In this paper, we derive an asymptotic formula for the sum \begin{equation*} \sum_{n \leq x} D(n + s(n)), \end{equation*} where \begin{equation*} D(n) = \frac{τ(n)}{2^{ω(n)}}. \end{equation*} Here, denotes the number of positive divisors of , and stands for the number of distinct prime factors of .
Paper Structure (3 sections, 2 theorems, 53 equations)

This paper contains 3 sections, 2 theorems, 53 equations.

Table of Contents

  1. Introduction
  2. Main Result
  3. Proof of Theorem 2.1

Key Result

Theorem 2.1

Let $x \geq 2$ be a real number. Then we have the asymptotic formula where $\varepsilon > 0$ is an arbitrary fixed constant, $\gamma$ denotes the Euler--Mascheroni constant, and the constants $C_1$ and $C_2$ are defined by with $P(s) = \prod_{p} \left(1 - \frac{1}{2p^s} + \frac{1}{2p^{2s}}\right)$.

Theorems & Definitions (3)

  • Theorem 2.1
  • Lemma 2.2
  • proof