Geodesic
Théorie des nombres
On the denominators of harmonic numbers. IV
Wu, Bing-Ling1 ; Yan, Xiao-Hui2 
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, Tome 360 (2022) no. G1, pp. 53-57.

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Let ℒ be the set of all positive integers n such that the denominator of 1+1/2+⋯+1/n is less than the least common multiple of 1,2,⋯,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.

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DOI : 10.5802/crmath.282
Classification : 11B05, 11B75
Mots-clés : harmonic numbers, least common multiples, upper asymptotic density

Wu, Bing-Ling 1 ; Yan, Xiao-Hui 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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Wu, Bing-Ling; Yan, Xiao-Hui. On the denominators of harmonic numbers. IV. Comptes Rendus. Mathématique, Tome 360 (2022) no. G1, pp. 53-57. doi : 10.5802/crmath.282. https://geodesic-test.mathdoc.fr/articles/10.5802/crmath.282/

[1] Boyd, David W. A p-adic study of the partial sums of the harmonic series, Exp. Math., Volume 3 (1994) no. 4, pp. 287-302 | DOI | MR | Zbl

[2] Eswarathasan, Arulappah; Levine, Eugene p-integral harmonic sums, Discrete Math., Volume 91 (1991) no. 3, pp. 249-257 | DOI | MR | Zbl

[3] Hardy, Godfrey H.; Wright, Edward M. An introduction to the theory of numbers, Oxford University Press, 1979

[4] Lang, Serge Introduction to transcendental numbers, Addison-Wesley Series in Mathematics, Addison-Wesley Publishing Group, 1966

[5] Sanna, Carlo On the p-adic valuation of harmonic numbers, J. Number Theory, Volume 166 (2016), pp. 41-46 | DOI | MR | Zbl

[6] Shiu, Peter The denominators of harmonic numbers (2016) (https://arxiv.org/abs/1607.02863v1)

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

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

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

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