On the denominators of harmonic numbers. IV
Let $\mathcal{L}$ be the set of all positive integers $n$ such that the denominator of $1+1/2+\cdots +1/n$ is less than the least common multiple of $1, 2, \dots , n$. In this paper, under a certain assumption on linear independence, we prove that the set $\mathcal{L}$ has the upper asymptotic densi...
| Main Authors: | , |
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| Format: | Article |
| Language: | English |
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Académie des sciences
2022-01-01
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| Series: | Comptes Rendus. Mathématique |
| Subjects: | |
| Online Access: | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.282/ |
| _version_ | 1797651571405750272 |
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| author | Wu, Bing-Ling Yan, Xiao-Hui |
| author_facet | Wu, Bing-Ling Yan, Xiao-Hui |
| author_sort | Wu, Bing-Ling |
| collection | DOAJ |
| description | Let $\mathcal{L}$ be the set of all positive integers $n$ such that the denominator of $1+1/2+\cdots +1/n$ is less than the least common multiple of $1, 2, \dots , n$. In this paper, under a certain assumption on linear independence, we prove that the set $\mathcal{L}$ has the upper asymptotic density $1$. The assumption follows from Schanuel’s conjecture. |
| first_indexed | 2024-03-11T16:16:43Z |
| format | Article |
| id | doaj.art-dbe0a600aa714edf8842b3675171c813 |
| institution | Directory Open Access Journal |
| issn | 1778-3569 |
| language | English |
| last_indexed | 2024-03-11T16:16:43Z |
| publishDate | 2022-01-01 |
| publisher | Académie des sciences |
| record_format | Article |
| series | Comptes Rendus. Mathématique |
| spelling | doaj.art-dbe0a600aa714edf8842b3675171c8132023-10-24T14:19:32ZengAcadémie des sciencesComptes Rendus. Mathématique1778-35692022-01-01360G1535710.5802/crmath.28210.5802/crmath.282On the denominators of harmonic numbers. IVWu, Bing-Ling0Yan, Xiao-Hui1School of Science, Nanjing University of Posts and Telecommunications, Nanjing 210023, P. R. ChinaSchool of Mathematics and Statistics, Anhui Normal University, Wuhu 241002, P. R. ChinaLet $\mathcal{L}$ be the set of all positive integers $n$ such that the denominator of $1+1/2+\cdots +1/n$ is less than the least common multiple of $1, 2, \dots , n$. In this paper, under a certain assumption on linear independence, we prove that the set $\mathcal{L}$ has the upper asymptotic density $1$. The assumption follows from Schanuel’s conjecture.https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.282/harmonic numbersleast common multiplesupper asymptotic density |
| spellingShingle | Wu, Bing-Ling Yan, Xiao-Hui On the denominators of harmonic numbers. IV Comptes Rendus. Mathématique harmonic numbers least common multiples upper asymptotic density |
| title | On the denominators of harmonic numbers. IV |
| title_full | On the denominators of harmonic numbers. IV |
| title_fullStr | On the denominators of harmonic numbers. IV |
| title_full_unstemmed | On the denominators of harmonic numbers. IV |
| title_short | On the denominators of harmonic numbers. IV |
| title_sort | on the denominators of harmonic numbers iv |
| topic | harmonic numbers least common multiples upper asymptotic density |
| url | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.282/ |
| work_keys_str_mv | AT wubingling onthedenominatorsofharmonicnumbersiv AT yanxiaohui onthedenominatorsofharmonicnumbersiv |