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$${\left(\sum _{n\in A\cap [1,x)}\frac{1}{n}\right)}^{-2}\sum _{\begin{array}{c}a,b\in A\cap (1,x]\\ a<b\end{array}}\frac{1}{\mathrm{lcm}(a,b)}\to \infty ?$$

AMS 11 · solved (literature)

theorem erdos_442 : answer(False) ↔ ∀ (A : Set ℕ),
    Tendsto (fun (x : ℝ) =>
      1 / x.maxLogOne.maxLogOne * ∑ n ∈ (A ∩ Icc 1 ⌊x⌋₊ : Set ℕ), (1 : ℝ) / n) atTop atTop →
    Tendsto (fun (x : ℝ) => 1 / (∑ n ∈ (A ∩ Icc 1 ⌊x⌋₊ : Set ℕ), (1 : ℝ) / n) ^ 2 *
      ∑ nm ∈ A.bddProdUpper x, (1 : ℝ) / nm.1.lcm nm.2) atTop atTop