FormalConjectures/ErdosProblems/517.lean

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L4DEL
Lean4 Domain Explanation Language

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

2 declarations, 2 contain sorry, 0 declared as axioms

Erdős Problem 517

References:

Imports (1)
Imported by 1 file
  1. theorem Erdos517.erdos_517 If f(z) = ∑ aₖzⁿₖ is an entire function (with aₖ ≠ 0 for all k) such that nₖ / k → ∞, is it true that f assumes every value infinitely often? sorry line 38 source

    Source documentation

    If f(z) = ∑ aₖzⁿₖ is an entire function (with aₖ ≠ 0 for all k) such that nₖ / k → ∞, is it true that f assumes every value infinitely often?

    depends on sorry what this means

    Statement uses: Iff, True, Complex, Nat, HasFabryGaps, Ne, OfNat.ofNat, HasSum, HMul.hMul, HPow.hPow, SummationFilter.unconditional, Set.Infinite, Set.ofPred, Eq

    Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous

    35 instances, coercions and auxiliary constants left out.

    Tactic trace: 1 steps

    Open to load the checked before-and-after goal states.

    theorem erdos_517 : answer(sorry) ↔ ∀ {f : ℂ → ℂ} {n : ℕ → ℕ} (hn : HasFabryGaps n)
    {a : ℕ → ℂ} (ha : ∀ k, a k ≠ 0) (hf : ∀ z, HasSum (fun k => a k * z ^ n k) (f z)) (z : ℂ),
    {x : ℂ | f x = z}.Infinite := by
…

    Mentions: HasFabryGaps

  2. theorem Erdos517.erdos_517.variants.fejer If f(z) = ∑ aₖzⁿₖ is an entire function (with aₖ ≠ 0 for all k) such that ∑ 1 / nₖ < ∞, then f assumes every value infinitely often sorry line 46 source

    Source documentation

    If f(z) = ∑ aₖzⁿₖ is an entire function (with aₖ ≠ 0 for all k) such that ∑ 1 / nₖ < ∞, then f assumes every value infinitely often. This theorem is proved in [Bi28].

    depends on sorry what this means

    Statement uses: Complex, Nat, HasFejerGaps, Ne, OfNat.ofNat, HasSum, HMul.hMul, HPow.hPow, SummationFilter.unconditional, Set.Infinite, Set.ofPred, Eq

    Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous

    35 instances, coercions and auxiliary constants left out.

    Tactic trace: 1 steps

    Open to load the checked before-and-after goal states.

    theorem erdos_517.variants.fejer {f : ℂ → ℂ} {n : ℕ → ℕ} (hn : HasFejerGaps n) {a : ℕ → ℂ}
    (ha : ∀ k, a k ≠ 0) (hf : ∀ z, HasSum (fun k => a k * z ^ n k) (f z)) (z : ℂ) :
    {x : ℂ | f x = z}.Infinite := by
…

    Mentions: HasFejerGaps