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Erdős problem 317

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Is there some constant c>0c>0 such that for every n≥1n\geq 1 there exists some δk∈{−1,0,1}\delta_k\in \{-1,0,1\} for 1≤k≤n1\leq k\leq n with 0<∣∑1≤k≤nδkk∣<c2n?0< \left\lvert \sum_{1\leq k\leq n}\frac{\delta_k}{k}\right\rvert < \frac{c}{2^n}?

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FormalConjectures/ErdosProblems/

317.lean

Retained formal statement3 of 4

Is it true that for sufficiently large nn, for any δk∈{−1,0,1}\delta_k\in \{-1,0,1\}, ∣∑1≤k≤nδkk∣>1[1,…,n]\left\lvert \sum_{1\leq k\leq n}\frac{\delta_k}{k}\right\rvert > \frac{1}{[1,\ldots,n]} whenever the left-hand side is not zero?

FormalConjectures/ErdosProblems/317.leanErdos317.erdos_317.variants.claim24 linesExact file
sorry ↔  ∀ᶠ (n : ℕ) in Filter.atTop,    ∀ (δ : Fin n → ℚ),      δ '' Set.univ ⊆ {-1, 0, 1} → |∑ k, δ k / (↑↑k + 1)| ≠ 0 → |∑ k, δ k / (↑↑k + 1)| > 1 / ↑((Finset.Icc 1 n).lcm id)
OpenStatement only, no proof