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 statement1 of 3

Is it true that $\limsup M_n = \infty$ ?

This is Problem 4.1 in [Ha74] where it is attributed to Erdős.

The weaker conjecture that $\limsup M_n=\infty$ was proved by Wagner [Wa80], who show that there is some $c>0$ with $M_n>(\log n)^c$ infinitely often.

FormalConjectures/ErdosProblems/119.leanErdos119.erdos_119.parts.i1 lineExact file

1True ↔ ∀ (z : ℕ → ℂ), (∀ (i : ℕ), ‖z i‖ = 1) → Filter.limsup (fun n => ↑(Erdos119.M z n)) Filter.atTop = ⊤

SolvedStatement only, no proof

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