A note on the $q$-adic valuation of $σ_k(n)$
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866910231630446592 |
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| author | Bordellès, Olivier |
| author_facet | Bordellès, Olivier |
| contents | In this note, we obtain an exact formula for the $q$-adic valuation of $σ_k(n)$ where $q$ is an odd prime, allowing us to derive an explicit upper bound which is asymptotically better than the previous bound obtained by Zhao when $n$ is large and $k \geqslant q-2$. The key parts are played by the LTE lemma and the use of cyclotomic polynomials. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_17872 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A note on the $q$-adic valuation of $σ_k(n)$ Bordellès, Olivier Number Theory 11A25, 11A99 In this note, we obtain an exact formula for the $q$-adic valuation of $σ_k(n)$ where $q$ is an odd prime, allowing us to derive an explicit upper bound which is asymptotically better than the previous bound obtained by Zhao when $n$ is large and $k \geqslant q-2$. The key parts are played by the LTE lemma and the use of cyclotomic polynomials. |
| title | A note on the $q$-adic valuation of $σ_k(n)$ |
| topic | Number Theory 11A25, 11A99 |
| url | https://arxiv.org/abs/2605.17872 |