The sum of reciprocals of least common multiples
Let \((a_i)_{i\ge 1}\) be a strictly increasing sequence of positive integers. In the paper under review, the authors consider the sums \[S:= \sum_{i=1}^n {\mathrm{lcm}} (a_i,\ldots, a_{i+r-1})^{-1}\] under the condition \({\mathrm{lcm}} (a_1, \ldots, a_{r-1}) \le a_r\), and determine the best upper bound for \(S\), depending uniformly only on \(r\) and \(n\). They also characterize the terms \(a_1,\ldots,a_{n+r-1}\) of the given sequence such that the best upper bound is attained. These results extend earlier related results by \textit{D. Borwein} [Can. Math. Bull. 21, 117--118 (1978; Zbl 0376.10003)] and \textit{G. Qian} [C. R., Math., Acad. Sci. Paris 355, No. 11, 1127--1132 (2017; Zbl 1401.11006)]. For Part II, see [\textit{J. Zhao} et al., Appl. Math. Comput. 399, Article ID 126003, 12 p. (2021; Zbl 07423476)].
scientific article; zbMATH DE number 7479504