On the Asymptotic Behavior of a Multiplicative Arithmetic Function Related to the Divisor Function Over Perfect Squares Integers Generated by Shifting
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866915813607342080 |
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| author | Mihoub, Bouderbala |
| author_facet | Mihoub, Bouderbala |
| contents | 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 $n$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_20808 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On the Asymptotic Behavior of a Multiplicative Arithmetic Function Related to the Divisor Function Over Perfect Squares Integers Generated by Shifting Mihoub, Bouderbala Number Theory 11A25, 11N37 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 $n$. |
| title | On the Asymptotic Behavior of a Multiplicative Arithmetic Function Related to the Divisor Function Over Perfect Squares Integers Generated by Shifting |
| topic | Number Theory 11A25, 11N37 |
| url | https://arxiv.org/abs/2602.20808 |