Dense sets of natural numbers with unusually large least common multiples
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866918194749374464 |
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| author | Tao, Terence |
| author_facet | Tao, Terence |
| contents | For any constant $C_0>0$, we construct a set $A \subset {\mathbb N}$ such that one has $$ \sum_{n \in A: n \leq x} \frac{1}{n} = \exp\left(\left(\frac{C_0}{2}+o(1)\right) (\log\log x)^{1/2} \log\log\log x \right)$$ and $$ \sum_{n,m \in A: n, m \leq x} \frac{1}{\operatorname{lcm}(n,m)} \ll_{C_0} \left(\sum_{n \in A: n \leq x} \frac{1}{n}\right)^2$$ as $x \to \infty$, with the growth rate given here optimal up to the dependence on $C_0$. This answers in the negative a question of Erdős and Graham, and also clarifies the nature of certain ``mostly coprime'' sets studied by Bergelson and Richter. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_04226 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Dense sets of natural numbers with unusually large least common multiples Tao, Terence Number Theory 11N56 For any constant $C_0>0$, we construct a set $A \subset {\mathbb N}$ such that one has $$ \sum_{n \in A: n \leq x} \frac{1}{n} = \exp\left(\left(\frac{C_0}{2}+o(1)\right) (\log\log x)^{1/2} \log\log\log x \right)$$ and $$ \sum_{n,m \in A: n, m \leq x} \frac{1}{\operatorname{lcm}(n,m)} \ll_{C_0} \left(\sum_{n \in A: n \leq x} \frac{1}{n}\right)^2$$ as $x \to \infty$, with the growth rate given here optimal up to the dependence on $C_0$. This answers in the negative a question of Erdős and Graham, and also clarifies the nature of certain ``mostly coprime'' sets studied by Bergelson and Richter. |
| title | Dense sets of natural numbers with unusually large least common multiples |
| topic | Number Theory 11N56 |
| url | https://arxiv.org/abs/2407.04226 |