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

TL;DR

This paper derives an asymptotic formula for a sum involving a multiplicative function related to divisors over integers shifted to the nearest perfect square, revealing its long-term behavior.

Contribution

It provides the first asymptotic analysis of a divisor-related multiplicative function evaluated at integers shifted to perfect squares.

Findings

Established an explicit asymptotic formula for the sum over n ≤ x.

Analyzed the behavior of the function D(n) in the context of perfect square shifts.

Enhanced understanding of divisor functions in shifted integer sequences.

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{\tau(n)}{2^{\omega(n)}}. \end{equation*} Here, $\tau(n)$ denotes the number of positive divisors of $n$, and $\omega(n)$ stands for the number of distinct prime factors of $n$.