On the asymptotic density of $k$-tuples of positive integers with pairwise non-coprime components
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866916510419648512 |
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| author | Tóth, László |
| author_facet | Tóth, László |
| contents | We use the convolution method for arithmetic functions of several variables to deduce an asymptotic formula for the number of $k$-tuples of positive integers with components which are pairwise non-coprime and $\le x$. More generally, we obtain asymptotic formulas on the number of $k$-tuples $(n_1,\ldots,n_k)\in {\Bbb N}^k$ such that at least $r$ pairs $(n_i,n_j)$, respectively exactly $r$ pairs are coprime. Our results answer the questions raised by Moree (2005, 2014), and generalize and refine related results obtained by Heyman (2014) and Hu (2014). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_08433 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the asymptotic density of $k$-tuples of positive integers with pairwise non-coprime components Tóth, László Number Theory Combinatorics 11A25, 11N25, 11N37, 05A15, 05C07 We use the convolution method for arithmetic functions of several variables to deduce an asymptotic formula for the number of $k$-tuples of positive integers with components which are pairwise non-coprime and $\le x$. More generally, we obtain asymptotic formulas on the number of $k$-tuples $(n_1,\ldots,n_k)\in {\Bbb N}^k$ such that at least $r$ pairs $(n_i,n_j)$, respectively exactly $r$ pairs are coprime. Our results answer the questions raised by Moree (2005, 2014), and generalize and refine related results obtained by Heyman (2014) and Hu (2014). |
| title | On the asymptotic density of $k$-tuples of positive integers with pairwise non-coprime components |
| topic | Number Theory Combinatorics 11A25, 11N25, 11N37, 05A15, 05C07 |
| url | https://arxiv.org/abs/2402.08433 |