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| Main Author: | |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2406.12283 |
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| _version_ | 1866916290621341696 |
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| author | Wang, Biao |
| author_facet | Wang, Biao |
| contents | Let $d^{(k)}(n)$ be the $k$-free divisor function for integer $k\ge2$. Let $a$ be a nonzero integer. In this paper, we establish an asymptotic formula \begin{equation*}
\sum_{p\leq x} d^{(k)}(p-a) =b_k \cdot x+O\left(\frac{x}{\log x}\right) \end{equation*} related to the Titchmarsh divisor problem, where $b_k$ is a positive constant dependent on $k$ and $a$. For the proof, we apply a result of Felix and show a general asymptotic formula for a class of arithmetic functions including the unitary divisor function, $k$-free divisor function and the proper Pillai's function. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_12283 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A generalization of the Titchmarsh divisor problem Wang, Biao Number Theory Let $d^{(k)}(n)$ be the $k$-free divisor function for integer $k\ge2$. Let $a$ be a nonzero integer. In this paper, we establish an asymptotic formula \begin{equation*} \sum_{p\leq x} d^{(k)}(p-a) =b_k \cdot x+O\left(\frac{x}{\log x}\right) \end{equation*} related to the Titchmarsh divisor problem, where $b_k$ is a positive constant dependent on $k$ and $a$. For the proof, we apply a result of Felix and show a general asymptotic formula for a class of arithmetic functions including the unitary divisor function, $k$-free divisor function and the proper Pillai's function. |
| title | A generalization of the Titchmarsh divisor problem |
| topic | Number Theory |
| url | https://arxiv.org/abs/2406.12283 |