Erdős problem 119 · problems.science

For unit-modulus complex numbers $z_i$ , let $p_n(z)=\prod_{i\le n}(z-z_i)$ and $M_n=\max_{|z|=1}|p_n(z)|$ . Erdős's prize question: is there $c>0$ with $\sum_{k\le n} M_k > n^{1+c}$ ?

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4 retained statements2415f78e850a

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FormalConjectures/ErdosProblems/

119.lean

Retained formal statement3 of 3

Is it true that there exists $c > 0$ such that, for all large $n$ , $\sum_{k \leq n} M_k > n^{1 + c}$ ?

The $100 prize was offered for the third question in [Er97f]. This was resolved by GPT 5.6 and Korsky (see the proof claims), who proved that $\sum_{k\leq n}M_k \gg \frac{n^{5/4}}{\sqrt{\log n}}$ (and hence for infinitely many $n$ we have $M_n> n^{1/4-o(1)}$ ).

FormalConjectures/ErdosProblems/119.leanErdos119.erdos_119.parts.iii4 linesExact file

1True ↔2  ∀ (z : ℕ → ℂ),3    (∀ (i : ℕ), ‖z i‖ = 1) →4      ∃ c, ∃ (_ : c > 0), ∀ᶠ (n : ℕ) in Filter.atTop, ∑ k ∈ Finset.range n, Erdos119.M z k > ↑n ^ (1 + c)

SolvedStatement only, no proof

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