Weighted Versions of the Arithmetic-Mean-Geometric Mean Inequality and Zaremba's Function
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2023
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| _version_ | 1866913605021073408 |
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| author | McCormack, Tim Zelinsky, Joshua |
| author_facet | McCormack, Tim Zelinsky, Joshua |
| contents | We use the weighted version of the arithmetic-mean-geometric-mean inequality to motivate new results about Zaremba's function, $z(n) = \sum_{d|n} \frac{\log d}{d}$. We investigate record-setting values for $z(n)$ and the related function $v(n) = \frac{z(n)}{\log τ(n)}$ where $τ(n)$ is the number of divisors of $n$. We show that $v(n)$ takes on a maximum value and we give a list of all record-setting values for$v(n)$. Closely connected inequalities motivate the study of numbers which are pseudoperfect in a strong sense. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2312_11661 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Weighted Versions of the Arithmetic-Mean-Geometric Mean Inequality and Zaremba's Function McCormack, Tim Zelinsky, Joshua Number Theory 11A05, 11A25, 26D15 We use the weighted version of the arithmetic-mean-geometric-mean inequality to motivate new results about Zaremba's function, $z(n) = \sum_{d|n} \frac{\log d}{d}$. We investigate record-setting values for $z(n)$ and the related function $v(n) = \frac{z(n)}{\log τ(n)}$ where $τ(n)$ is the number of divisors of $n$. We show that $v(n)$ takes on a maximum value and we give a list of all record-setting values for$v(n)$. Closely connected inequalities motivate the study of numbers which are pseudoperfect in a strong sense. |
| title | Weighted Versions of the Arithmetic-Mean-Geometric Mean Inequality and Zaremba's Function |
| topic | Number Theory 11A05, 11A25, 26D15 |
| url | https://arxiv.org/abs/2312.11661 |