Resolution of Erdős' problems about unimodularity
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arXiv
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866916569918996480 |
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| author | Cambie, Stijn |
| author_facet | Cambie, Stijn |
| contents | Letting $δ_1(n,m)$ be the density of the set of integers with exactly one divisor in $(n,m)$, Erdős wondered if $δ_1(n,m)$ is unimodular for fixed $n$.
We prove this is false in general, as the sequence $(δ_1(n,m))$ has superpolynomially many local extrema. However, we confirm unimodality in the single case for which it occurs; $n = 1$.
We also solve the question on unimodality of the density of integers whose $k^{th}$ prime is $p$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_10333 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Resolution of Erdős' problems about unimodularity Cambie, Stijn Number Theory Combinatorics Probability 11A41, 11A51, 11K36 Letting $δ_1(n,m)$ be the density of the set of integers with exactly one divisor in $(n,m)$, Erdős wondered if $δ_1(n,m)$ is unimodular for fixed $n$. We prove this is false in general, as the sequence $(δ_1(n,m))$ has superpolynomially many local extrema. However, we confirm unimodality in the single case for which it occurs; $n = 1$. We also solve the question on unimodality of the density of integers whose $k^{th}$ prime is $p$. |
| title | Resolution of Erdős' problems about unimodularity |
| topic | Number Theory Combinatorics Probability 11A41, 11A51, 11K36 |
| url | https://arxiv.org/abs/2501.10333 |