897.lean`

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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

5 declarations, 5 contain sorry, 0 declared as axioms

Erdős Problem 897

References:

erdosproblems.com/897
[Ar25] Archivara Math Research Agent, An Additive Counterexample: Erdős Problem 897 (2025)
[ArWu25] Aristotle, operated mostly by L. Wu, Lean formalisation of Erdős problem 897 (2025)
[Wi70] E. Wirsing, A characterization of $\log n$ as an additive arithmetic function. Symposia Math. (1970), 45-57.
[Wi81] E. Wirsing, Additive and completely additive functions with restricted growth. Recent progress in analytic number theory, Vol. 2 (Durham, 1979), 231--280 (1981).

Imports (1)

FormalConjecturesUtil

theorem Erdos897.erdos_897.parts.i sorry line 48 source
depends on sorry what this means
Statement uses: Iff, False, Nat, Real, GT.gt, OfNat.ofNat, Nat.Coprime, Eq, HMul.hMul, HAdd.hAdd, EReal, Filter.limsup, Prod, Erdos897.erdos_897.parts.i.match_1, HDiv.hDiv, HPow.hPow, Real.log, Min.min, Filter, Filter.atTop and 5 more
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
60 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem Erdos897.erdos_897.parts.ii sorry line 64 source
depends on sorry what this means
Statement uses: Iff, False, Nat, Real, GT.gt, OfNat.ofNat, Nat.Coprime, Eq, HMul.hMul, HAdd.hAdd, EReal, Filter.limsup, Prod, Erdos897.erdos_897.parts.i.match_1, HDiv.hDiv, HPow.hPow, Real.log, Min.min, Filter, Filter.atTop and 4 more
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
52 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem Erdos897.erdos_897.variants.log_growth sorry line 77 source
depends on sorry what this means
Statement uses: Nat, Real, GT.gt, OfNat.ofNat, Nat.Coprime, Eq, HMul.hMul, HAdd.hAdd, LE.le, abs, HSub.hSub, Exists, And, Asymptotics.IsBigO, Filter.atTop, Real.log
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
40 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem Erdos897.erdos_897.variants.parts.i Let $f(n)$ be an additive function (so that $f(ab)=f(a)+f(b)$ if $(a,b)=1$) such that $\limsup_{p,k} f(p^k) / \log(p^k) = ∞$ and $f(p^k) = f(p)$ or $f(p^k) = kf(p)$ sorry line 96 source
Source documentation
Let $f(n)$ be an additive function (so that $f(ab)=f(a)+f(b)$ if $(a,b)=1$) such that $\limsup_{p,k} f(p^k) / \log(p^k) = ∞$ and $f(p^k) = f(p)$ or $f(p^k) = kf(p)$. Is it true that $\limsup_n (f(n+1)−f(n))/ \log n = ∞$?
The known counterexample does not satisfy either of these extra hypotheses, so this variant remains open.

depends on sorry what this means
Statement uses: Iff, True, Nat, Real, GT.gt, OfNat.ofNat, Nat.Coprime, Eq, HMul.hMul, HAdd.hAdd, EReal, Filter.limsup, Prod, Erdos897.erdos_897.parts.i.match_1, HDiv.hDiv, HPow.hPow, Real.log, Min.min, Filter, Filter.atTop and 6 more
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
62 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_897.variants.parts.i : answer(sorry) ↔ ∀ (f : ℕ → ℝ),
    (∀ᵉ (a > 0) (b > 0), a.Coprime b → f (a * b) = f a + f b) →
    ((Filter.atTop ⊓ Filter.principal {(p, k) : ℕ × ℕ | p.Prime}).limsup
      (fun (p, k) => (f (p^k) / (p^k : ℝ).log : EReal)) = ⊤) →
    (∀ k p, p.Prime → f (p^k) = f p) ∨ (∀ (k p : ℕ), p.Prime → f (p^k) = k*f p) →
    Filter.atTop.limsup (fun (n : ℕ) => ((f (n+1) - f n) / (n : ℝ).log : EReal)) = ⊤ := by
…
```
theorem Erdos897.erdos_897.variants.parts.ii Let $f(n)$ be an additive function (so that $f(ab)=f(a)+f(b)$ if $(a,b)=1$) such that $\limsup_{p,k} f(p^k) / \log(p^k) = ∞$ and $f(p^k) = f(p)$ or $f(p^k) = kf(p)$ sorry line 114 source
Source documentation
Let $f(n)$ be an additive function (so that $f(ab)=f(a)+f(b)$ if $(a,b)=1$) such that $\limsup_{p,k} f(p^k) / \log(p^k) = ∞$ and $f(p^k) = f(p)$ or $f(p^k) = kf(p)$. Is it true that $\limsup_n f(n+1)/f(n) = ∞$?
The known counterexample does not satisfy either of these extra hypotheses, so this variant remains open.

depends on sorry what this means
Statement uses: Iff, True, Nat, Real, GT.gt, OfNat.ofNat, Nat.Coprime, Eq, HMul.hMul, HAdd.hAdd, EReal, Filter.limsup, Prod, Erdos897.erdos_897.parts.i.match_1, HDiv.hDiv, HPow.hPow, Real.log, Min.min, Filter, Filter.atTop and 5 more
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
54 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_897.variants.parts.ii : answer(sorry) ↔ ∀ (f : ℕ → ℝ),
    (∀ᵉ (a > 0) (b > 0), a.Coprime b → f (a * b) = f a + f b) →
    ((Filter.atTop ⊓ Filter.principal {(p, k) : ℕ × ℕ | p.Prime}).limsup
      (fun (p, k) => (f (p^k) / (p^k : ℝ).log : EReal)) = ⊤) →
    (∀ k p, p.Prime → f (p^k) = f p) ∨ (∀ (k p : ℕ), p.Prime → f (p^k) = k*f p) →
    Filter.atTop.limsup (fun (n : ℕ) => (f (n+1) / f n : EReal)) = ⊤ := by
…
```