Soit une suite strictement croissante d'entiers positifs ( pour ). En 1978, Borwein a montré que, pour tout entier positif n, on a , avec égalité si et seulement si pour . Soit un entier. Dans cette Note, nous étudions les sommes et nous montrons qu'elles sont majorées, pour tout entier positif r, par une constante dépendant de r et n. De plus, pour tout entier , nous caractérisons aussi les suites pour lesquelles l'égalité est vérifiée.
Let be a strictly increasing sequence of positive integers ( if ). In 1978, Borwein showed that for any positive integer n, we have , with equality occurring if and only if for . Let be an integer. In this paper, we investigate the sum and show that for any positive integer n, where is a constant depending on r and n. Further, for any integer , we also give a characterization of the sequence such that the equality holds.
Accepté le :
Publié le :
Guoyou Qian 1
@article{CRMATH_2017__355_11_1127_0, author = {Guoyou Qian}, title = {On the sum of reciprocals of least common multiples}, journal = {Comptes Rendus. Math\'ematique}, pages = {1127--1132}, publisher = {Elsevier}, volume = {355}, number = {11}, year = {2017}, doi = {10.1016/j.crma.2017.10.015}, language = {en}, }
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] Generalization of an inequality of Heilbronn and Rohrbach, Bull. Amer. Math. Soc., Volume 54 (1948), pp. 681-684
[2] A sum of reciprocals of least common multiples, Can. Math. Bull., Volume 20 (1978), pp. 117-118
[3] Memoire sur les nombres premiers, J. Math. Pures Appl., Volume 17 (1852), pp. 366-390
[4] Inequalities involving least common multiple and other arithmetical functions, Indag. Math., Volume 20 (1958), pp. 5-15
[5] 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] Nontrivial lower bounds for the least common multiple of some finite sequences of integers, J. Number Theory, Volume 125 (2007), pp. 393-411
[7] On the average asymptotic behavior of a certain type of sequences of integers, Integers, Volume 9 (2009), pp. 555-567
[8] On the product of the primes, Can. Math. Bull., Volume 15 (1972), pp. 33-37
[9] On an inequality in the elementary theory of numbers, Math. Proc. Camb. Philos. Soc., Volume 33 (1937), pp. 207-209
[10] 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] 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] New lower bounds for the least common multiple of polynomial sequences, Number Theory, Volume 175 (2017), pp. 191-199
[13] Asymptotic improvements of lower bounds for the least common multiples of arithmetic progressions, Can. Math. Bull., Volume 57 (2014), pp. 551-561
[14] 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.
Commentaires - Politique