On the sum of reciprocals of least common multiples

Qian, Guoyou

doi:10.1016/j.crma.2017.10.015

Number theory

On the sum of reciprocals of least common multiples
[Sur les sommes des inverses de plus petits communs multiples]

Présenté par : the Editorial Board

Qian, Guoyou ¹

¹ Mathematical College, Sichuan University, Chengdu 610064, PR China

Comptes Rendus. Mathématique, Tome 355 (2017) no. 11, pp. 1127-1132.

Résumé
Abstract

Soit ${a_{i}}_{i = 1}^{\infty}$ une suite strictement croissante d'entiers positifs ( $a_{i} < a_{j}$ pour $i < j$ ). En 1978, Borwein a montré que, pour tout entier positif n, on a $\sum_{i = 1}^{n} \frac{1}{ppcm (a_{i}, a_{i + 1})} \leq 1 - \frac{1}{2^{n}}$ , avec égalité si et seulement si $a_{i} = 2^{i - 1}$ pour $1 \leq i \leq n + 1$ . Soit $3 \leq r \leq 7$ un entier. Dans cette Note, nous étudions les sommes $\sum_{i = 1}^{n} \frac{1}{ppcm (a_{i}, \dots, a_{i + r - 1})}$ et nous montrons qu'elles sont majorées, pour tout entier positif r, par une constante $U_{r} (n)$ dépendant de r et n. De plus, pour tout entier $n \geq 2$ , nous caractérisons aussi les suites ${a_{i}}_{i = 1}^{\infty}$ pour lesquelles l'égalité $\sum_{i = 1}^{n} \frac{1}{ppcm (a_{i}, \dots, a_{i + r - 1})} = U_{r} (n)$ est vérifiée.

Let ${a_{i}}_{i = 1}^{\infty}$ be a strictly increasing sequence of positive integers ( $a_{i} < a_{j}$ if $i < j$ ). In 1978, Borwein showed that for any positive integer n, we have $\sum_{i = 1}^{n} \frac{1}{lcm (a_{i}, a_{i + 1})} \leq 1 - \frac{1}{2^{n}}$ , with equality occurring if and only if $a_{i} = 2^{i - 1}$ for $1 \leq i \leq n + 1$ . Let $3 \leq r \leq 7$ be an integer. In this paper, we investigate the sum $\sum_{i = 1}^{n} \frac{1}{lcm (a_{i}, . . ., a_{i + r - 1})}$ and show that $\sum_{i = 1}^{n} \frac{1}{lcm (a_{i}, . . ., a_{i + r - 1})} \leq U_{r} (n)$ for any positive integer n, where $U_{r} (n)$ is a constant depending on r and n. Further, for any integer $n \geq 2$ , we also give a characterization of the sequence ${a_{i}}_{i = 1}^{\infty}$ such that the equality $\sum_{i = 1}^{n} \frac{1}{lcm (a_{i}, . . ., a_{i + r - 1})} = U_{r} (n)$ holds.

Reçu le : 2017-04-25
Accepté le : 2017-10-26
Publié le : 2017-11-01

DOI : 10.1016/j.crma.2017.10.015

Affiliations des auteurs :

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.