Superior Highly Composite Numbers and the Explicit Upper Bound of Generalized Divisor Functions
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866918214220382208 |
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| author | Teo, Lee-Peng |
| author_facet | Teo, Lee-Peng |
| contents | For $k\geq 2$, we give a detailed exposition of the superior $k$-highly composite numbers. We then consider the function \[f_k(n)=\frac{\log d_k(n)\log\log n}{\log k\log n},\quad n\geq 3\] which has a maximum value $λ(k)$ at a superior $k$-highly composite number. We develop an efficient algorithm to compute $λ(k)$ and the positive integer $N_{\max}(k)$ where $f_k$ achieves the value $λ(k)$. The results for $2\leq k\leq 100$ are tabled. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_06764 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Superior Highly Composite Numbers and the Explicit Upper Bound of Generalized Divisor Functions Teo, Lee-Peng Number Theory For $k\geq 2$, we give a detailed exposition of the superior $k$-highly composite numbers. We then consider the function \[f_k(n)=\frac{\log d_k(n)\log\log n}{\log k\log n},\quad n\geq 3\] which has a maximum value $λ(k)$ at a superior $k$-highly composite number. We develop an efficient algorithm to compute $λ(k)$ and the positive integer $N_{\max}(k)$ where $f_k$ achieves the value $λ(k)$. The results for $2\leq k\leq 100$ are tabled. |
| title | Superior Highly Composite Numbers and the Explicit Upper Bound of Generalized Divisor Functions |
| topic | Number Theory |
| url | https://arxiv.org/abs/2508.06764 |