Arithmetic functions at factorial arguments
Fuente:
arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2023
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| Acceso en línea: | |
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| _version_ | 1866910461248667648 |
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| author | De Koninck, Jean-Marie Verreault, William |
| author_facet | De Koninck, Jean-Marie Verreault, William |
| contents | For various arithmetic functions $f:\mathbb{N} \to \mathbb{R}$, the behavior of $f(n!)$ and that of $\sum_{n\le N} f(n!)$ can be intriguing. For instance, for some functions $f$, we have ${f(n!)=\sum_{k\le n}f(k)}$, for others, we have ${f(n!)=\sum_{p\le n}f(p)}$ (where the sum runs over all the primes $p\le n$). Also, for some $f$, their minimum order coincides with $\lim_{n\to \infty}f(n!)$, for others, it is their maximum order that does so. Here, we elucidate such phenomena and more generally, we embark on a study of $f(n!)$ and of $\sum_{n\le N}f(n!)$ for a wide variety of arithmetical functions $f$. In particular, letting $d(n)$ and $σ(n)$ stand respectively for the number of positive divisors of $n$ and the sum of the positive divisors of $n$, we obtain new accurate asymptotic expansions for $d(n!)$ and $σ(n!)$. Furthermore, setting $ρ_1(n):=\max\{d\mid n:d\le \sqrt n\}$ and observing that no one has yet obtained an asymptotic value for $\sum_{n\le N} ρ_1(n)$ as $N\to \infty$, we show how one can obtain the asymptotic value of $\sum_{n\le N} ρ_1(n!)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2308_09761 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Arithmetic functions at factorial arguments De Koninck, Jean-Marie Verreault, William Number Theory 11A25 (Primary) 11N37, 11B65 (Secondary) For various arithmetic functions $f:\mathbb{N} \to \mathbb{R}$, the behavior of $f(n!)$ and that of $\sum_{n\le N} f(n!)$ can be intriguing. For instance, for some functions $f$, we have ${f(n!)=\sum_{k\le n}f(k)}$, for others, we have ${f(n!)=\sum_{p\le n}f(p)}$ (where the sum runs over all the primes $p\le n$). Also, for some $f$, their minimum order coincides with $\lim_{n\to \infty}f(n!)$, for others, it is their maximum order that does so. Here, we elucidate such phenomena and more generally, we embark on a study of $f(n!)$ and of $\sum_{n\le N}f(n!)$ for a wide variety of arithmetical functions $f$. In particular, letting $d(n)$ and $σ(n)$ stand respectively for the number of positive divisors of $n$ and the sum of the positive divisors of $n$, we obtain new accurate asymptotic expansions for $d(n!)$ and $σ(n!)$. Furthermore, setting $ρ_1(n):=\max\{d\mid n:d\le \sqrt n\}$ and observing that no one has yet obtained an asymptotic value for $\sum_{n\le N} ρ_1(n)$ as $N\to \infty$, we show how one can obtain the asymptotic value of $\sum_{n\le N} ρ_1(n!)$. |
| title | Arithmetic functions at factorial arguments |
| topic | Number Theory 11A25 (Primary) 11N37, 11B65 (Secondary) |
| url | https://arxiv.org/abs/2308.09761 |