Comptes Rendus
Number theory
On the sum of reciprocals of least common multiples
[Sur les sommes des inverses de plus petits communs multiples]
Comptes Rendus. Mathématique, Volume 355 (2017) no. 11, pp. 1127-1132.

Soit {ai}i=1 une suite strictement croissante d'entiers positifs (ai<aj pour i<j). En 1978, Borwein a montré que, pour tout entier positif n, on a i=1n1ppcm(ai,ai+1)112n, avec égalité si et seulement si ai=2i1 pour 1in+1. Soit 3r7 un entier. Dans cette Note, nous étudions les sommes i=1n1ppcm(ai,,ai+r1) et nous montrons qu'elles sont majorées, pour tout entier positif r, par une constante Ur(n) dépendant de r et n. De plus, pour tout entier n2, nous caractérisons aussi les suites {ai}i=1 pour lesquelles l'égalité i=1n1ppcm(ai,,ai+r1)=Ur(n) est vérifiée.

Let {ai}i=1 be a strictly increasing sequence of positive integers (ai<aj if i<j). In 1978, Borwein showed that for any positive integer n, we have i=1n1lcm(ai,ai+1)112n, with equality occurring if and only if ai=2i1 for 1in+1. Let 3r7 be an integer. In this paper, we investigate the sum i=1n1lcm(ai,...,ai+r1) and show that i=1n1lcm(ai,...,ai+r1)Ur(n) for any positive integer n, where Ur(n) is a constant depending on r and n. Further, for any integer n2, we also give a characterization of the sequence {ai}i=1 such that the equality i=1n1lcm(ai,...,ai+r1)=Ur(n) holds.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2017.10.015

Guoyou Qian 1

1 Mathematical College, Sichuan University, Chengdu 610064, PR China
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Guoyou Qian. On the sum of reciprocals of least common multiples. Comptes Rendus. Mathématique, Volume 355 (2017) no. 11, pp. 1127-1132. doi : 10.1016/j.crma.2017.10.015. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2017.10.015/

[1] F.A. Behrend Generalization of an inequality of Heilbronn and Rohrbach, Bull. Amer. Math. Soc., Volume 54 (1948), pp. 681-684

[2] D. Borwein A sum of reciprocals of least common multiples, Can. Math. Bull., Volume 20 (1978), pp. 117-118

[3] P.L. Chebyshev Memoire sur les nombres premiers, J. Math. Pures Appl., Volume 17 (1852), pp. 366-390

[4] J.G. van der Corput Inequalities involving least common multiple and other arithmetical functions, Indag. Math., Volume 20 (1958), pp. 5-15

[5] B. Farhi Minoration non triviales du plus petit commun multiple de certaines suites finies d'entiers, C. R. Acad. Sci. Paris, Ser. I, Volume 341 (2005), pp. 469-474

[6] B. Farhi Nontrivial lower bounds for the least common multiple of some finite sequences of integers, J. Number Theory, Volume 125 (2007), pp. 393-411

[7] B. Farhi On the average asymptotic behavior of a certain type of sequences of integers, Integers, Volume 9 (2009), pp. 555-567

[8] D. Hanson On the product of the primes, Can. Math. Bull., Volume 15 (1972), pp. 33-37

[9] H.A. Heilbronn On an inequality in the elementary theory of numbers, Math. Proc. Camb. Philos. Soc., Volume 33 (1937), pp. 207-209

[10] S. Hong; W. Feng Lower bounds for the least common multiple of finite arithmetic progressions, C. R. Acad. Sci. Paris, Ser. I, Volume 343 (2006), pp. 695-698

[11] S. Hong; Y. Luo; G. Qian; C. Wang Uniform lower bound for the least common multiple of a polynomial sequence, C. R. Acad. Sci. Paris, Ser. I, Volume 351 (2013), pp. 781-785

[12] S. Hong; G. Qian New lower bounds for the least common multiple of polynomial sequences, Number Theory, Volume 175 (2017), pp. 191-199

[13] D. Kane; S. Kominers Asymptotic improvements of lower bounds for the least common multiples of arithmetic progressions, Can. Math. Bull., Volume 57 (2014), pp. 551-561

[14] M. Nair On Chebyshev-type inequalities for primes, Amer. Math. Mon., Volume 89 (1982), pp. 126-129

Cité par Sources :

The research was supported partially by National Science Foundation of China Grant #11501387, by Young Teacher's Science Foundation of Sichuan University Grant #2015SCU11043, and by International Visiting program for Excellent Young Scholars of Sichuan University.

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