On a family of arithmetic series related to the Möbius function
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866910039687561216 |
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| author | Tenenbaum, Gérald |
| author_facet | Tenenbaum, Gérald |
| contents | Let $P^-(n)$ denote the smallest prime factor of a natural integer $n>1$. Furthermore let $μ$ and $ω$ denote respectively the Möbius function and the number of distinct prime factors function. We show that, given any set ${\scr P}$ of prime numbers with a natural density, we have $\sum_{P^-(n)\in \scr P}μ(n)ω(n)/n=0$ and provide a effective estimate for the rate of convergence. This extends a recent result of Alladi and Johnson, who considered the case when ${\scr P}$ is an arithmetic progression. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_02754 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On a family of arithmetic series related to the Möbius function Tenenbaum, Gérald Number Theory 11N 37, 11N25, 11N56 Let $P^-(n)$ denote the smallest prime factor of a natural integer $n>1$. Furthermore let $μ$ and $ω$ denote respectively the Möbius function and the number of distinct prime factors function. We show that, given any set ${\scr P}$ of prime numbers with a natural density, we have $\sum_{P^-(n)\in \scr P}μ(n)ω(n)/n=0$ and provide a effective estimate for the rate of convergence. This extends a recent result of Alladi and Johnson, who considered the case when ${\scr P}$ is an arithmetic progression. |
| title | On a family of arithmetic series related to the Möbius function |
| topic | Number Theory 11N 37, 11N25, 11N56 |
| url | https://arxiv.org/abs/2409.02754 |