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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 statement3 of 5

Cambie [Ca25] has shown that, for fixed nn, the sequence δ1(n,m)\delta_1(n,m) has superpolynomially many local maxima mm.

FormalConjectures/ErdosProblems/692.leanErdos692.erdos_692.variants.cambie_local_maxima3 linesExact file
∀ (k : ℕ) (δ : ℕ → ℕ → ℝ),  (∀ (a b : ℕ), Erdos692.IsDelta₁ a b (δ a b)) →    ∀ᶠ (n : ℕ) in Filter.atTop, ↑n ^ k ≤ ↑{m | n + 1 < m ∧ δ n (m - 1) ≤ δ n m ∧ δ n (m + 1) ≤ δ n m}.ncard
SolvedStatement only, no proof