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Erdős Problem #119

Analysis · posed by Paul Erdős, 1957 · solved

2 comments

Statement

For unit-modulus complex numbers ziz_i, let pn(z)=∏i≤n(z−zi)p_n(z)=\prod_{i\le n}(z-z_i) and Mn=max⁡∣z∣=1∣pn(z)∣M_n=\max_{|z|=1}|p_n(z)|. Erdős's prize question: is there c>0c>0 with ∑k≤nMk>n1+c\sum_{k\le n} M_k > n^{1+c}?

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No person has examined this. Nothing here has been checked at all. say whether it holds →

  1. proof attempt · #1

    Samuel Korsky, using GPT-5.6

    That credit came with the record as it was imported. No ProbXiv account is credited for this work, and nobody has answered for it here.

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    ai assisted
    — a person led the work and used a model along the way.

    GPT-5.6, with Samuel Korsky, resolved Erdős's prize question, proving ∑k≤nMk≫n5/4/log⁡n\sum_{k\le n} M_k \gg n^{5/4}/\sqrt{\log n} (hence Mn>n1/4−o(1)M_n > n^{1/4-o(1)} infinitely often).

  2. Recorded elsewhere on #1 · not checked here

    recorded: correctVibeMathed site check

    scope Reproduction by the VibeMathed site

    Marked solved on erdosproblems.com; carried an Erdős prize of USD 100. Resolved via a proof claim by GPT-5.6 and Samuel Korsky; not formally Lean-verified.

    Repeated from the source; nothing was checked here.

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