On the Generalised Divisor Problem
Fuente:
arXiv
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| Autor principal: | |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866908845006127104 |
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| author | Tudzi, Sebastian |
| author_facet | Tudzi, Sebastian |
| contents | In this paper, we apply the Dirichlet convolution method to \begin{equation*}
T_{k}(x)=\sum_{n \leq x} d_{k}(n), \end{equation*} for $k\ge 3$, where $d_{k}(n)$ is the number of ways to represent $n$ as a product of $k$ positive integer factors. We prove that for $k=3$, the error term $|Δ_3(x)|< 2.968x^{2/3}\log^{1/3}x$ for all $x\ge 2$. This improves the best-known explicit result established by Bordell{è}s for all $x\ge 2$. We extend this for all $k>3$ and obtain an explicit error term of the form $Δ_{k}(x)=O\left(x^{\frac{k-1}{k}}(\log x)^{\frac{(k-1)(k-2)}{2k}}\right)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_21823 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the Generalised Divisor Problem Tudzi, Sebastian Number Theory In this paper, we apply the Dirichlet convolution method to \begin{equation*} T_{k}(x)=\sum_{n \leq x} d_{k}(n), \end{equation*} for $k\ge 3$, where $d_{k}(n)$ is the number of ways to represent $n$ as a product of $k$ positive integer factors. We prove that for $k=3$, the error term $|Δ_3(x)|< 2.968x^{2/3}\log^{1/3}x$ for all $x\ge 2$. This improves the best-known explicit result established by Bordell{è}s for all $x\ge 2$. We extend this for all $k>3$ and obtain an explicit error term of the form $Δ_{k}(x)=O\left(x^{\frac{k-1}{k}}(\log x)^{\frac{(k-1)(k-2)}{2k}}\right)$. |
| title | On the Generalised Divisor Problem |
| topic | Number Theory |
| url | https://arxiv.org/abs/2506.21823 |