On the natural density of integers $n$ for which $σ(kn+r_1) >σ(kn+r_2)$
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2023
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| _version_ | 1866917333915664384 |
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| author | Luo, Xin-qi Ren, Chen-kai |
| author_facet | Luo, Xin-qi Ren, Chen-kai |
| contents | For any positive integer $n$, let $σ(n)=\sum_{d\mid n} d$. In 2020, M. Kobayashi and T. Trudgian showed that the natural density of positive integers n with $σ(kn+r_1) \geq σ(kn+r_2)$ is between 0.053 and 0.055. In this paper, we extend their result. For integers $k>r_1>r_2\geq 0,$ we provide an estimate on the natural density of positive integers $n$ for which $σ(kn+r_1) > σ(kn+r_2)$. We also calculate some special cases with certain $k,r_1$ and $r_2$. We also compute explicit bounds for specific $k,r_1,r_2$ to illustrate the variation of density. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2311_00295 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On the natural density of integers $n$ for which $σ(kn+r_1) >σ(kn+r_2)$ Luo, Xin-qi Ren, Chen-kai Number Theory For any positive integer $n$, let $σ(n)=\sum_{d\mid n} d$. In 2020, M. Kobayashi and T. Trudgian showed that the natural density of positive integers n with $σ(kn+r_1) \geq σ(kn+r_2)$ is between 0.053 and 0.055. In this paper, we extend their result. For integers $k>r_1>r_2\geq 0,$ we provide an estimate on the natural density of positive integers $n$ for which $σ(kn+r_1) > σ(kn+r_2)$. We also calculate some special cases with certain $k,r_1$ and $r_2$. We also compute explicit bounds for specific $k,r_1,r_2$ to illustrate the variation of density. |
| title | On the natural density of integers $n$ for which $σ(kn+r_1) >σ(kn+r_2)$ |
| topic | Number Theory |
| url | https://arxiv.org/abs/2311.00295 |