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Is it true that if $A\subseteq\mathbb{N}$ is such that \[\frac{1}{\log\log x}\sum_{n\in A\cap [1,x)}\frac{1}{n}\to \infty\] then \[\left(\sum_{n\in A\cap [1,x)}\frac{1}{n}\right)^{-2} \sum_{\substack{a,b\in A\cap (1,x]\\ a<b}}\frac{1}{\mathrm{lcm}(a,b)}\to \infty?\]
Tao [Ta24b] has shown this is false: there exists $A\subset\mathbb{N}$ such that \[\sum_{n\in A\cap [1,x)}\gg \exp((\tfrac{1}{2}+o(1))\sqrt{\log\log x}\log\log\log x)\] and \[\left(\sum_{n\in A\cap [1,x)}\frac{1}{n}\right)^{-2} \sum_{\substack{a,b\in A\cap (1,x]\\ a<b}}\frac{1}{\mathrm{lcm}(a,b)}\ll 1.\] Moreover, Tao shows this is the best possible result, in that if $\sum_{n\in A\cap [1,x)}\frac{1}{n}$ grows faster than $\exp(O(\sqrt{\log\log x}\log\log\log x))$ then \[\left(\sum_{n\in A\cap [1,x)}\frac{1}{n}\right)^{-2} \sum_{\substack{a,b\in A\cap (1,x]\\ a<b}}\frac{1}{\mathrm{lcm}(a,b)}\to \infty.\]