The exact upper bound for the sum of reciprocals of least common multiples

摘要

Let r be a positive number with r≥2 and let A={ai}i=1∞ be an arbitrarily given strictly increasing sequence of positive integers. Let SrA:=∑i=1∞1[ai,ai+1,⋯,ai+r−1]. In 1978, Borwein obtained S2A≤1 with equality occurring if and only if ai=2i−1 for i≥1. Qian and Zhao et al. obtained exact upper bounds for SrA as 3≤r≤7 and 8≤r≤11 respectively in 2017 and 2019. In this paper, we give several methods to obtain the upper bounds for S12A and S13A with explicit sequences which reach the corresponding upper bounds. We propose a conjecture that the exact upper bounds for SrA are τ(h)−r+2h for all r≥2, where h is a highly composite number and τ(h) denotes the number of divisors of h. In addition, we offer some sequences that reach the exact upper bounds.