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Erdős Problem #1039

Analysis · posed by 1958 · solved

2 comments

Statement

For f(z)=∏i=1n(z−zi)f(z) = \prod_{i=1}^n (z - z_i) with all ∣zi∣≤1|z_i| \le 1, let ρ(f)\rho(f) be the radius of the largest disc contained in {z:∣f(z)∣<1}\{z : |f(z)| < 1\}. Is ρ(f)≫1/n\rho(f) \gg 1/n? The worst case is now known to be Θ(1/n)\Theta(1/n), with the explicit bound ρ(f)≥(log⁡2)/n\rho(f) \ge (\log 2)/n.

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No person has examined this. Everything below was judged by machines. say whether it holds →

  1. proof attempt · #1

    GPT-5.5 Pro, Codex 5.5

    No person is named on this work: the record names only the tool it came from, and no ProbXiv account is credited for it.

    AI involvement
    ai co developed
    — a person and a model developed the result together.

    The bounds were developed with GPT-5.5 Pro and Codex 5.5.

    order of magnitude determined; the exact asymptotic constant remains open

  2. Machine-checked by Lean on #1 · not a person

    lean: correctLean

    scope Lean formalization of the result

    Lean-checked and expert-vouched; official record updated.

    Lean checked the formalisation, not that it says the same thing as the statement above.

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Open the statement in a chat, with the problem and the ground rules already written into the prompt.

open in Claude →open in ChatGPT →

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