516.lean`

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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, 3 contain sorry, 0 declared as axioms

Erdős Problem 516

References:

erdosproblems.com/516
[Fu63] Fuchs, W. H. J., Proof of a conjecture of G. Pólya concerning gap series. Illinois J. Math. (1963), 661--667.
[Ko65] Kövari, Thomas, A gap-theorem for entire functions of infinite order. Michigan Math. J. (1965), 133--140.
[Ki26] Kitamura, Kenta, A Lean counterexample to the Fejér-gap variant of Erdős Problem 516. GitHub (2026).

Imports (1)

FormalConjecturesUtil

def Erdos516.OfFiniteOrder An entire function f is said to be of finite order if there exist numbers c, a ≥ 0 such that for all z, ‖f z‖ ≤ c * rexp (‖z‖ ^ a) line 41 source
Source documentation
An entire function f is said to be of finite order if there exist numbers c, a ≥ 0 such that for all z, ‖f z‖ ≤ c * rexp (‖z‖ ^ a).

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, NormedSpace, Complex
Proof uses: And, Differentiable, Exists, Real, GE.ge, OfNat.ofNat, Nat, LE.le, Norm.norm, HMul.hMul, Real.exp, HPow.hPow
23 instances, coercions and auxiliary constants left out.
```
def OfFiniteOrder {E F: Type*} [NormedAddCommGroup E] [NormedSpace ℂ E]
    [NormedAddCommGroup F] [NormedSpace ℂ F] (f : E → F) : Prop :=
…
```
def Erdos516.ratio line 45 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Real, Complex
Proof uses: HDiv.hDiv, Real.log, iInf, Subtype, Eq, Norm.norm, iSup
7 instances, coercions and auxiliary constants left out.
theorem Erdos516.erdos_516 sorry line 52 source
depends on sorry what this means
Statement uses: Complex, Nat, HasFabryGaps, Ne, OfNat.ofNat, HasSum, HMul.hMul, HPow.hPow, SummationFilter.unconditional, Erdos516.OfFiniteOrder, Eq, Real, Filter.limsup, Erdos516.ratio, Filter.atTop
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
53 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem Erdos516.erdos_516.variants.limsup_ratio_eq_one Let f = ∑ aₖzⁿₖ be an entire function with strictly increasing exponents such that nₖ > k (log k) ^ (2 + c) sorry line 63 source
Source documentation
Let f = ∑ aₖzⁿₖ be an entire function with strictly increasing exponents such that nₖ > k (log k) ^ (2 + c). Then limsup (fun r => ratio r f) atTop = 1. This is proved in [Ko65]. Without StrictMono n, repeated exponents let nonzero coefficients cancel, and f = 0 satisfies the other hypotheses.

depends on sorry what this means
Statement uses: Complex, Nat, StrictMono, Exists, Real, And, GT.gt, OfNat.ofNat, HMul.hMul, HPow.hPow, Real.log, HAdd.hAdd, Ne, HasSum, SummationFilter.unconditional, Eq, Filter.limsup, Erdos516.ratio, Filter.atTop
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
70 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_516.variants.limsup_ratio_eq_one {f : ℂ → ℂ} {n : ℕ → ℕ} (hmono : StrictMono n)
    (hn : ∃ c > (0 : ℝ), ∀ k, n k > k * log k ^ (2 + c)) {a : ℕ → ℂ} (ha : ∀ n, a n ≠ 0)
    (hfn : ∀ z, HasSum (fun k => a k * z ^ n k) (f z)) :
    limsup (fun r => ratio r f) atTop = 1 := by
…
```
Mentions: c, Erdos516.ratio
theorem Erdos516.erdos_516.variants.limsup_ratio_eq_one_of_hasFejerGaps sorry line 76 source
depends on sorry what this means
Statement uses: Iff, False, Complex, Nat, HasFejerGaps, Ne, OfNat.ofNat, HasSum, HMul.hMul, HPow.hPow, SummationFilter.unconditional, Eq, Real, Filter.limsup, Erdos516.ratio, Filter.atTop
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
45 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.