Number theory
On the sum of reciprocals of least common multiples
[Sur les sommes des inverses de plus petits communs multiples]
Comptes Rendus. Mathématique, Tome 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
Qian, Guoyou 1

1 Mathematical College, Sichuan University, Chengdu 610064, PR China
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Qian, Guoyou. On the sum of reciprocals of least common multiples. Comptes Rendus. Mathématique, Tome 355 (2017) no. 11, pp. 1127-1132. doi : 10.1016/j.crma.2017.10.015. http://www.numdam.org/articles/10.1016/j.crma.2017.10.015/

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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.