239.lean`

Explanation legend Lean4 Domain Explanation Language (L4DEL)

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L4DEL: Lean4 Domain Explanation Language

Annotation provenance. This overlay contains 10 curated AI-written annotations and 7190 mechanical source-derived annotations. They have not been human-reviewed as explanations; treat them as unverified and not mathematical authority. Retained generation history: The 71 original curated annotations were produced in a Claude Sonnet 5.5 session at medium effort; Claude Opus 5.5 is recorded in that session as the advisor model, not the direct author. The bulk source-derived expansion was generated mechanically from pinned source comments by tooling developed in a GPT-5.6 Sol session at ultra reasoning; GPT-5.6 Sol did not originate the quoted source prose. On 2026-10-17 the mechanically generated source-comment annotations of the Formal Conjectures, Carleson, FLT, PFR, CompFiles, Mathematics in Lean and Theorem Proving in Lean overlays were regenerated by the same generator after three corrections made in a Claude Sonnet 5.5 session: titles no longer cut a sentence inside mathematics or end on a dangling word, workflow markers are removed from the quoted prose, and declaration names follow the scanner's namespace resolution. The boilerplate step line was dropped, and comments that were only a row of dashes no longer yield annotations. No new prose was written and no human review is claimed.

Credit: The upstream authors and contributors retain credit for the project and mathematics. lean4-explainer only renders their pinned source and adds the separately disclosed explanation layer.

Project sources: Pinned project documentation, Formal Conjectures paper, arXiv v1

1 declarations, 1 contain sorry, 0 declared as axioms

Erdős Problem 239

References:

erdosproblems.com/239
[Ha68] Halász, G., Über die Mittelwerte multiplikativer zahlentheoretischer Funktionen. Acta Math. Acad. Sci. Hungar. (1968), 365-403.
[Wi67] Wirsing, E., Das asymptotische Verhalten von Summen über multiplikative Funktionen. Acta Math. Acad. Sei. Hung. (1967), 411-467.

Imports (1)

FormalConjecturesUtil

theorem Erdos239.erdos_239 Let $f:\mathbb{N}\to \{-1,1\}$ be a multiplicative function sorry line 45 source
Source documentation
Let $f:\mathbb{N}\to \{-1,1\}$ be a multiplicative function. Is it true that $$ \lim_{N\to \infty}\frac{1}{N}\sum_{n\leq N}f(n)$$ always exists?
The answer is yes, as proved by Wirsing [Wi67], and generalised by Halász [Ha68].

depends on sorry what this means
Statement uses: Iff, True, Nat, Real, And, GE.ge, OfNat.ofNat, Or, Eq, Neg.neg, Nat.Coprime, HMul.hMul, Exists, Filter.Tendsto, HDiv.hDiv, Finset.sum, Finset.Icc, Filter.atTop, nhds
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
38 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_239 :
    answer(True) ↔ ∀ f : ℕ → ℝ,
    (∀ n ≥ 1, f n = 1 ∨ f n = -1) ∧
    (∀ m n, m.Coprime n → f (m * n) = f m * f n) ∧
    f 1 = 1 →
    ∃ L, Tendsto (fun N ↦ (∑ n ∈ Finset.Icc 1 N, f n) / N) atTop (𝓝 L) := by
…
```