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Erdős problem 906

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Is there an entire non-zero function f:C→Cf:\mathbb{C}\to \mathbb{C} such that, for any infinite sequence n1<n2<⋯n_1<n_2<\cdots, the set {z:f(nk)(z)=0 for some k≥1}\{ z: f^{(n_k)}(z)=0 \textrm{ for some }k\geq 1\} is everywhere dense? The literal question is trivial for polynomials, so the claims address the transcendental entire case, in the affirmative.

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FormalConjectures/ErdosProblems/

906.lean

Retained formal statement1 of 1

Does there exists an entire non-zero transcendental function f : ℂ → ℂ such that for any sequence n₀ < n₁ < ..., { z | ∃ k, iteratedDeriv (n k) f z = 0 } is dense.

FormalConjectures/ErdosProblems/906.leanErdos906.erdos_9064 linesExact file
True ↔  ∃ f,    Transcendental (Polynomial ℂ) f ∧      Differentiable ℂ f ∧ ∀ (n : ℕ → ℕ), StrictMono n → Dense {z | ∃ k, iteratedDeriv (n k) f z = 0}
OpenStatement only, no proof