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Erdős ProblemsNumber theory

Erdős Problem 317

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}?

Number theoryAMS 11

Mathematical statement

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}?

Statement source: Erdős Problems statement material

Statement terms: Source-specific

Source-specific terms. Therefore does not assert reuse rights beyond attributed display.

Statement artifacts, not proofs

These records expose exact Lean propositions and statement-only wrappers. Defining a proposition does not supply a proof of it. A placeholder-bearing target also contains no proof. Elaboration checks syntax and types; it does not certify that a formalization perfectly captures every nuance of the informal problem.

Pinned Lean formulation 1

erdos_317

Canonical source
Complete statement target, proof intentionally absentLean 4
theorem erdos_317 : answer(sorry) ↔    ∃ c > 0, ∀ n ≥ 1, ∃ δ : Fin n → ℚ,      Set.range δ ⊆ {-1, 0, 1} ∧      letI lhs : ℝ := |∑ k, (δ k) / (k + 1)|      0 < lhs ∧ lhs < c / 2^n := by  sorry
Statement source
Formal Conjectures
Lean version
v4.27.0
Placeholder
Present; no proof artifact
Source evidence
Pinned source index
Fidelity review
Community formulation

References

Pinned Formal Conjectures sourceformalizationerdosproblems.com/317background