On the sum of $Δ_{k}(n)$ in the Piltz divisor problem for $k=3$ and $k=4$
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866915064011816960 |
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| author | Minamide, T. Makoto Tanigawa, Yoshio Watt, Nigel |
| author_facet | Minamide, T. Makoto Tanigawa, Yoshio Watt, Nigel |
| contents | Let $Δ_{k}(x)$ be the error term in the classical asymptotic formula for the sum $\sum_{n\leq x}d_{k}(n)$, where $d_{k}(n)$ is the number of ways $n$ can be written as a product of $k$ factors. We study the analytic properties of the Dirichlet series $\sum_{n=1}^{\infty}Δ_{k}(n)n^{-s}$ and use Perron's formula to estimate the sums $\sum_{n\leq x}Δ_{3}(n)$ and $\sum_{n\leq x}Δ_{4}(n)$ for large $x>0$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_11069 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the sum of $Δ_{k}(n)$ in the Piltz divisor problem for $k=3$ and $k=4$ Minamide, T. Makoto Tanigawa, Yoshio Watt, Nigel Number Theory 11N37, 11M06 Let $Δ_{k}(x)$ be the error term in the classical asymptotic formula for the sum $\sum_{n\leq x}d_{k}(n)$, where $d_{k}(n)$ is the number of ways $n$ can be written as a product of $k$ factors. We study the analytic properties of the Dirichlet series $\sum_{n=1}^{\infty}Δ_{k}(n)n^{-s}$ and use Perron's formula to estimate the sums $\sum_{n\leq x}Δ_{3}(n)$ and $\sum_{n\leq x}Δ_{4}(n)$ for large $x>0$. |
| title | On the sum of $Δ_{k}(n)$ in the Piltz divisor problem for $k=3$ and $k=4$ |
| topic | Number Theory 11N37, 11M06 |
| url | https://arxiv.org/abs/2412.11069 |