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Erdős ProblemsNumber theory

Erdős Problem 520

Let ff be a Rademacher multiplicative function. Does there exist some constant c>0c > 0 such that, almost surely, lim sup⁡N→∞∑m≤Nf(m)Nlog⁡log⁡N=c?\limsup_{N \to \infty} \frac{\sum_{m \leq N} f(m)}{\sqrt{N \log \log N}} = c?

Number theoryAMS 11AMS 60

Mathematical statement

Let ff be a Rademacher multiplicative function. Does there exist some constant c>0c > 0 such that, almost surely,

lim sup⁡N→∞∑m≤Nf(m)Nlog⁡log⁡N=c? \limsup_{N \to \infty} \frac{\sum_{m \leq N} f(m)}{\sqrt{N \log \log N}} = c?

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_520

Canonical source
Complete statement target, proof intentionally absentLean 4
theorem erdos_520 :    answer(sorry) ↔ ∃ c > 0, ∀ (Ω : Type) [MeasureSpace Ω] [IsProbabilityMeasure (ℙ : Measure Ω)]      (f : ℕ → Ω → ℝ), IsRademacherMultiplicative f →      ∀ᵐ ω, limsup (fun N ↦ ∑ m ≤ N, f m ω / sqrt (N * log (log N))) atTop = c := 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/520background