Monotone non-decreasing sequences of the Euler totient function
Fuente:
arXiv
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| Autor principal: | |
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| Formato: | Preprint |
| Publicado: |
2023
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| _version_ | 1866911828721795072 |
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| author | Tao, Terence |
| author_facet | Tao, Terence |
| contents | Let $M(x)$ denote the largest cardinality of a subset of $\{n \in \mathbf{N}: n \leq x\}$ on which the Euler totient function $φ(n)$ is non-decreasing. We show that $M(x) = (1+O(\frac{(\log\log x)^5}{\log x})) π(x)$ for all $x \geq 10$, answering questions of Erdős and Pollack--Pomerance--Treviño. A similar result is also obtained for the sum of divisors function $σ(n)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2309_02325 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Monotone non-decreasing sequences of the Euler totient function Tao, Terence Number Theory 11A25 Let $M(x)$ denote the largest cardinality of a subset of $\{n \in \mathbf{N}: n \leq x\}$ on which the Euler totient function $φ(n)$ is non-decreasing. We show that $M(x) = (1+O(\frac{(\log\log x)^5}{\log x})) π(x)$ for all $x \geq 10$, answering questions of Erdős and Pollack--Pomerance--Treviño. A similar result is also obtained for the sum of divisors function $σ(n)$. |
| title | Monotone non-decreasing sequences of the Euler totient function |
| topic | Number Theory 11A25 |
| url | https://arxiv.org/abs/2309.02325 |