950.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

8 declarations, 7 contain sorry, 0 declared as axioms

Erdős Problem 950

References:

erdosproblems.com/855
erdosproblems.com/950
[Er77c] Erdős, Paul, Problems and results on combinatorial number theory. III. Number theory day (Proc. Conf., Rockefeller Univ., New York, 1976) (1977), 43-72.
mathoverflow/508491

Imports (1)

FormalConjecturesUtil

def Erdos950.f Let $f(n) = \sum_{p<n}\frac{1}{n-p}$ line 41 source
Source documentation
Let $f(n) = \sum_{p<n}\frac{1}{n-p}$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Real
Proof uses: Finset.sum, Finset.filter, Prime, Finset.range, HDiv.hDiv, OfNat.ofNat, HSub.hSub
12 instances, coercions and auxiliary constants left out.
```
noncomputable def f (n : ℕ) : ℝ :=
…
```
theorem Erdos950.erdos_950.parts.i Is it true that $\liminf f(n)=1$? sorry line 48 source
Source documentation
Is it true that $\liminf f(n)=1$?

depends on sorry what this means
Statement uses: Iff, True, Eq, EReal, Filter.liminf, Nat, Erdos950.f, Filter.atTop, OfNat.ofNat
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
17 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_950.parts.i :
    answer(sorry) ↔ atTop.liminf (fun n : ℕ ↦ (f n : EReal)) = 1 := by
…
```
Mentions: Erdos950.f
theorem Erdos950.erdos_950.parts.ii Is it true that $\limsup f(n)=\infty$? sorry line 56 source
Source documentation
Is it true that $\limsup f(n)=\infty$?

depends on sorry what this means
Statement uses: Iff, True, Eq, EReal, Filter.limsup, Nat, Erdos950.f, Filter.atTop, Top.top
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat
15 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_950.parts.ii :
    answer(sorry) ↔ atTop.limsup (fun n : ℕ ↦ (f n : EReal)) = ⊤ := by
…
```
Mentions: Erdos950.f
theorem Erdos950.erdos_950.parts.iii Is it true that $f(n)=o(\log\log n)$ for all $n$? sorry line 64 source
Source documentation
Is it true that $f(n)=o(\log\log n)$ for all $n$?

depends on sorry what this means
Statement uses: Iff, True, Asymptotics.IsLittleO, Nat, Real, Filter.atTop, Erdos950.f, Real.log
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat
9 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_950.parts.iii :
    answer(sorry) ↔ f =o[atTop] (fun n : ℕ ↦ Real.log (Real.log n)) := by
…
```
Mentions: Erdos950.f
theorem Erdos950.erdos_950.variants.debruijn_erdos_turan This function was considered by de Bruijn, Erdős, and Turán, who showed that $\sum_{n<x}f(n)\sim \sum_{n<x}f(n)^2\sim x$ sorry line 74 source
Source documentation
This function was considered by de Bruijn, Erdős, and Turán, who showed that $\sum_{n<x}f(n)\sim \sum_{n<x}f(n)^2\sim x$. They gave no proofs, but a proof of the (harder) second claim is given by Gorodetsky here [mathoverflow/508491].

depends on sorry what this means
Statement uses: And, Asymptotics.IsEquivalent, Nat, Real, Filter.atTop, Finset.sum, Finset.range, Erdos950.f, HPow.hPow, OfNat.ofNat
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
28 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_950.variants.debruijn_erdos_turan :
    (fun x : ℕ ↦ ∑ n ∈ Finset.range x, f n) ~[atTop] (fun x : ℕ ↦ (x : ℝ)) ∧
    (fun x : ℕ ↦ ∑ n ∈ Finset.range x, (f n) ^ 2) ~[atTop] (fun x : ℕ ↦ (x : ℝ)) := by
…
```
Mentions: Erdos950.f
theorem Erdos950.erdos_950.variants.weaker_pi Documented declaration: weaker_pi sorry line 85 source
Source documentation
Erdős writes that a 'weaker conjecture which is perhaps not quite inaccessible' is that, for every $\epsilon>0$, if $x$ is sufficiently large there exists $y<x$ such that $\pi(x)< \pi(y)+\epsilon \pi(x-y)$. Compare this to [855].

depends on sorry what this means
Statement uses: Iff, True, Real, GT.gt, OfNat.ofNat, Filter.Eventually, Nat, Exists, And, LT.lt, Nat.primeCounting, HAdd.hAdd, HMul.hMul, HSub.hSub, Filter.atTop
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
27 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_950.variants.weaker_pi :
    answer(sorry) ↔ ∀ ε > 0, ∀ᶠ x in atTop, ∃ y < x,
      (π x : ℝ) < (π y : ℝ) + ε * (π (x - y) : ℝ) := by
…
```
theorem Erdos950.erdos_950.variants.weaker_pi_implies_f He notes that if $\pi(x)< \pi(y)+O\left(\frac{x-y}{\log x}\right)$ for all $y0$ then $f(n)\ll \log\log\log n$ sorry line 95 source
Source documentation
He notes that if $\pi(x)< \pi(y)+O\left(\frac{x-y}{\log x}\right)$ for all $y<x-(\log x)^C$ for some constant $C>0$ then $f(n)\ll \log\log\log n$.

L4DEL explanation
He notes that if $\pi(x)< \pi(y)+O\left(\frac{x-y}{\log x}\right)$ for all $y0$ then $f(n)\ll \log\log\log n$.
Sources and attribution: Pinned Lean source (1646ca16afd6)

depends on sorry what this means
Statement uses: Exists, Real, And, GT.gt, OfNat.ofNat, Filter.Eventually, Nat, LT.lt, HSub.hSub, HPow.hPow, Real.log, Nat.primeCounting, HAdd.hAdd, HMul.hMul, HDiv.hDiv, Filter.atTop, Asymptotics.IsBigO, Erdos950.f
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
37 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_950.variants.weaker_pi_implies_f
    (h : ∃ C > (0 : ℝ), ∃ K > (0 : ℝ), ∀ᶠ x : ℕ in atTop, ∀ y : ℕ,
      (y : ℝ) < x - (Real.log x) ^ C →
      (π x : ℝ) < (π y : ℝ) + K * ((x - y : ℝ) / Real.log x)) :
    f =O[atTop] fun n : ℕ ↦ Real.log (Real.log (Real.log n)) := by
…
```
Mentions: Erdos950.f
theorem Erdos950.erdos_950.variants.sum_primes The study of $f(p)$ is even harder, and Erdős could not prove that $\sum_{p<x}f(p)^2\sim \pi(x)$ sorry line 107 source
Source documentation
The study of $f(p)$ is even harder, and Erdős could not prove that $\sum_{p<x}f(p)^2\sim \pi(x)$.

depends on sorry what this means
Statement uses: Iff, True, Asymptotics.IsEquivalent, Nat, Real, Filter.atTop, Finset.sum, Finset.filter, Prime, Finset.range, HPow.hPow, Erdos950.f, OfNat.ofNat, Nat.primeCounting
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
32 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_950.variants.sum_primes :
    answer(sorry) ↔
      (fun x : ℕ ↦ ∑ p ∈ (Finset.range x).filter Prime, (f p) ^ 2) ~[atTop]
        fun x ↦ (π x : ℝ) := by
…
```
Mentions: Erdos950.f