On a multiplicative property of sequences of integers (Q1207377)

Let \(l_ n(x)\) denote the \(n\)-iterated log function so that e.g. \(l_ 2(x)=\log\log x\) and let \([a,b]\) be the least common multiple of the integers \(a\) and \(b\). Erdős and Graham asked whether it is true that if \(a_ 1< a_ 2<\cdots\) is a sequence of integers satisfying \({1\over l_ 2(x)}\sum_{a_ i< x}{1\over a_ i}\to\infty\) then \[ \left(\sum_{a_ i< x}{1\over a_ i}\right)^{-2}\sum_{1< a_ i< a_ j\leq x} {1\over [a_ i,a_ j]}\to \infty. \] The authors show, with ingenious arguments, that the hypothesis is far two weak and indeed one needs \[ \sum_{a_ i<x} {1\over a_ i} >\exp\bigl(f(x)(l_ 2(x))^{1/2} l_ 3(x)\bigr), \] where \(f(x)\to \infty\), for the conclusion to follow. This is one of two corollaries to their Theorem 1 (which is rather too technical to be stated here) and Theorem 2 shows that this corollary is best possible.