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| Main Authors: | , |
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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2507.23041 |
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| _version_ | 1866915789559300096 |
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| author | McNew, Nathan Setty, Jai |
| author_facet | McNew, Nathan Setty, Jai |
| contents | We investigate the densities of the sets of abundant numbers and of covering numbers, integers $n$ for which there exists a distinct covering system where every modulus divides $n$. We establish that the set $\mathcal{C}$ of covering numbers possesses a natural density $d(\mathcal{C})$ and prove that $0.103230 < d(\mathcal{C}) < 0.103398.$ Our approach adapts methods developed by Behrend and Deléglise for bounding the density of abundant numbers, by introducing a function $c(n)$ that measures how close an integer $n$ is to being a covering number with the property that $c(n) \leq h(n) = σ(n)/n$. However, computing $d(\mathcal{C})$ to three decimal digits requires some new ideas to simplify the computations. As a byproduct of our methods, we obtain significantly improved bounds for $d(\mathcal{A})$, the density of abundant numbers, namely $0.247619608 < d(\mathcal{A}) < 0.247619658$. We also show the count of primitive covering numbers up to $x$ is $O\left( x\exp\left(\left(-\tfrac{1}{2\sqrt{\log 2}} + ε\right)\sqrt{\log x} \log \log x\right)\right)$, which is substantially smaller than the corresponding bound for primitive abundant numbers. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_23041 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the densities of covering numbers and abundant numbers McNew, Nathan Setty, Jai Number Theory 11B05 We investigate the densities of the sets of abundant numbers and of covering numbers, integers $n$ for which there exists a distinct covering system where every modulus divides $n$. We establish that the set $\mathcal{C}$ of covering numbers possesses a natural density $d(\mathcal{C})$ and prove that $0.103230 < d(\mathcal{C}) < 0.103398.$ Our approach adapts methods developed by Behrend and Deléglise for bounding the density of abundant numbers, by introducing a function $c(n)$ that measures how close an integer $n$ is to being a covering number with the property that $c(n) \leq h(n) = σ(n)/n$. However, computing $d(\mathcal{C})$ to three decimal digits requires some new ideas to simplify the computations. As a byproduct of our methods, we obtain significantly improved bounds for $d(\mathcal{A})$, the density of abundant numbers, namely $0.247619608 < d(\mathcal{A}) < 0.247619658$. We also show the count of primitive covering numbers up to $x$ is $O\left( x\exp\left(\left(-\tfrac{1}{2\sqrt{\log 2}} + ε\right)\sqrt{\log x} \log \log x\right)\right)$, which is substantially smaller than the corresponding bound for primitive abundant numbers. |
| title | On the densities of covering numbers and abundant numbers |
| topic | Number Theory 11B05 |
| url | https://arxiv.org/abs/2507.23041 |