Möbius function and primes: an identity factory with applications
Fuente:
arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2023
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| Materias: | |
| Acceso en línea: | |
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| _version_ | 1866914245972590592 |
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| author | Ramaré, Olivier Alterman, Sebastian Zuniga |
| author_facet | Ramaré, Olivier Alterman, Sebastian Zuniga |
| contents | We investigate the sums
$\sum_{n\le X, (n,q)=1}\frac{μ(n)}{n^s}\log^k\left(\frac{X}{n}\right)$, where $k\in\{0,1\}$,
$s\in\mathbb{C}$, $\Re s>0$. Our goal is to obtain explicit asymptotic estimations for these quantities. To achieve this, we develop a broad framework of identities that we use to derive several applications. Building on similar principles, we also provide an appendix establishing the inequality $\sum_{n\le
X}Λ(n)/n\le \log X$, valid for any $X\geq 1$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2312_05138 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Möbius function and primes: an identity factory with applications Ramaré, Olivier Alterman, Sebastian Zuniga Number Theory 11A25, 11A41, 11N37, 11N56 We investigate the sums $\sum_{n\le X, (n,q)=1}\frac{μ(n)}{n^s}\log^k\left(\frac{X}{n}\right)$, where $k\in\{0,1\}$, $s\in\mathbb{C}$, $\Re s>0$. Our goal is to obtain explicit asymptotic estimations for these quantities. To achieve this, we develop a broad framework of identities that we use to derive several applications. Building on similar principles, we also provide an appendix establishing the inequality $\sum_{n\le X}Λ(n)/n\le \log X$, valid for any $X\geq 1$. |
| title | Möbius function and primes: an identity factory with applications |
| topic | Number Theory 11A25, 11A41, 11N37, 11N56 |
| url | https://arxiv.org/abs/2312.05138 |