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Erdős problem 692

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Let δ1(n,m)\delta_1(n,m) be the density of the set of integers with exactly one divisor in (n,m)(n,m). Is δ1(n,m)\delta_1(n,m) unimodular for m>n+1m>n+1 (i.e. increases until some mm then decreases thereafter)?

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5 retained statements36c6d0fef71a

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FormalConjectures/ErdosProblems/

692.lean

Retained formal statement5 of 5

Erdős proved that δ1(n,m)≪1(log⁡n)c\delta_1(n,m) \ll \frac{1}{(\log n)^c} for all mm, for some constant c>0c>0.

FormalConjectures/ErdosProblems/692.leanErdos692.erdos_692.variants.erdos_upper_bound3 linesExact file
∀ (δ : ℕ → ℕ → ℝ),  (∀ (a b : ℕ), Erdos692.IsDelta₁ a b (δ a b)) →    ∃ c > 0, ∃ C > 0, ∀ᶠ (n : ℕ) in Filter.atTop, ∀ (m : ℕ), δ n m ≤ C / Real.log ↑n ^ c
SolvedStatement only, no proof