Resolution of Erdős' problems about unimodularity
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| Title: | Resolution of Erdős' problems about unimodularity |
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| Authors: | Stijn Cambie |
| Source: | Journal of Number Theory. 280:271-277 |
| Publisher Information: | Elsevier BV, 2026. |
| Publication Year: | 2026 |
| Subject Terms: | 11A41, 11A51, 11K36, Probability (math.PR), FOS: Mathematics, Number Theory (math.NT), Combinatorics (math.CO) |
| Description: | 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 $p$. 5 pages |
| Document Type: | Article |
| Language: | English |
| ISSN: | 0022-314X |
| DOI: | 10.1016/j.jnt.2025.08.014 |
| DOI: | 10.48550/arxiv.2501.10333 |
| Rights: | Elsevier TDM CC BY |
| Accession Number: | edsair.doi.dedup.....6fd79322bbc6e4d2b4b48f45c9ebb676 |
| Database: | OpenAIRE |
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Nájsť tento článok vo Web of Science | Abstract: | 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 $p$.<br />5 pages |
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| ISSN: | 0022314X |
| DOI: | 10.1016/j.jnt.2025.08.014 |