FormalConjectures.ErdosProblems.«1196»

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import FormalConjecturesUtil

Erdős Problem 1196

Reference: erdosproblems.com/1196


namespace Erdos1196


open scoped Asymptotics

A set is primitive if no non-associated elements of the set divide each other.

def IsPrimitive {M : Type*} [CommMonoid M] (A : Set M) : Prop :=
  ∀ᵉ (x ∈ A) (y ∈ A), x ∣ y → Associated x y

Is it true that, for any $x$, if $A\subset [x,\infty)$ is a primitive set of integers (so that no distinct elements of $A$ divide each other) then$$\sum_{a\in A}\frac{1}{a\log a}< 1+o(1),$$where the $o(1)$ term $\to 0$ as $x\to \infty$?

@[category research solved, AMS 11, formal_proof using lean4 at
  "https://github.com/math-inc/Erdos1196/blob/02fba13be7487cc51315f68d8fa7ef277633d3c8/PrimitiveSetsAboveX/FormalConjecturesErdos1196.lean"]
theorem declaration uses 'sorry'erdos_1196 :
    answer(True) ↔ ∃ o : ℕ → ℝ, o =o[Filter.atTop] (1 : ℕ → ℝ) ∧ ∀ x > (0 : ℕ), ∀ A ⊆ Set.Ici x, IsPrimitive A →
       ∑' (a : A), (1 / ((a.val : ℝ).log * a)) < 1 + o x  := by⊢ True ↔
  ∃ o, o =o[Filter.atTop] 1 ∧ ∀ x > 0, ∀ A ⊆ Set.Ici x, IsPrimitive A → ∑' (a : ↑A), 1 / (Real.log ↑↑a * ↑↑a) < 1 + o x
  sorryAll goals completed! 🐙


end Erdos1196