On the denominators of harmonic numbers. IV

Bing-Ling Wu; Xiao-Hui Yan

doi:10.5802/crmath.282

Théorie des nombres

On the denominators of harmonic numbers. IV

Bing-Ling Wu ¹ ; Xiao-Hui Yan ²

¹ School of Science, Nanjing University of Posts and Telecommunications, Nanjing 210023, P. R. China
² School of Mathematics and Statistics, Anhui Normal University, Wuhu 241002, P. R. China

Comptes Rendus. Mathématique, Volume 360 (2022), pp. 53-57.

Résumé

Let $ℒ$ be the set of all positive integers $n$ such that the denominator of $1 + 1 / 2 + \dots + 1 / n$ 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 $ℒ$ has the upper asymptotic density $1$ . The assumption follows from Schanuel’s conjecture.

Reçu le : 2021-08-11
Révisé le : 2021-08-25
Accepté le : 2021-10-12
Publié le : 2022-01-26

DOI : 10.5802/crmath.282

Classification : 11B05, 11B75
Mots clés : harmonic numbers, least common multiples, upper asymptotic density

Affiliations des auteurs :

Bing-Ling Wu ¹ ; Xiao-Hui Yan ²

¹ School of Science, Nanjing University of Posts and Telecommunications, Nanjing 210023, P. R. China
² School of Mathematics and Statistics, Anhui Normal University, Wuhu 241002, P. R. China

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     title = {On the denominators of harmonic numbers. {IV}},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {53--57},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {360},
     year = {2022},
     doi = {10.5802/crmath.282},
     language = {en},
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Bing-Ling Wu; Xiao-Hui Yan. On the denominators of harmonic numbers. IV. Comptes Rendus. Mathématique, Volume 360 (2022), pp. 53-57. doi : 10.5802/crmath.282. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.282/

Bibliographie
Cité par

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[7] Bing-Ling Wu; Yong-Gao Chen On certain properties of harmonic numbers, J. Number Theory, Volume 175 (2017), pp. 66-86 | MR | Zbl

[8] Bing-Ling Wu; Yong-Gao Chen On the denominators of harmonic numbers, C. R. Math. Acad. Sci. Paris, Volume 356 (2018) no. 2, pp. 129-132 | MR | Zbl

[9] Bing-Ling Wu; Yong-Gao Chen On the denominators of harmonic numbers. II, J. Number Theory, Volume 200 (2019), pp. 397-406 | MR | Zbl

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