Let $z_{i}$ be an infinite sequence of complex numbers such that $∣ z_{i} ∣ = 1$ for all $i \geq 1$ , and for $n \geq 1$ let $p_{n} (z) = i \leq n \prod (z - z_{i}) .$ Let $M_{n} = max_{∣ z ∣ = 1} ∣ p_{n} (z)∣$ . Is it true that $lim sup M_{n} = \infty$ ?Is it true that there exists $c > 0$ such that for infinitely many $n$ we have $M_{n} > n^{c}$ ?Is it true that there exists $c > 0$ such that, for all large $n$ , $k \leq n \sum M_{k} > n^{1 + c} ?$

Worked, still open.

analysis · open · prize $100 · formalized (Lean) · 0 attempts

machinery: unit-circle-polynomials,max-modulus-lower-bound,Cesaro-average-growth,Erdos-product-norm,analysis-polynomials,extremal-analysis

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vela registry pull vfr_37aec80d874a0239

vela reproduce examples/erdos-problems

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unverified AI candidates (2)

gpt-erdos · GPT-5.2 Pro + Deep Research · unverified

These questions are known (they are listed as an Erdős problem in Hayman’s problem list). The current state is:

llm-hunter · gpt pro 5.2 · unverified

1 LLM attack(s) recorded (gpt pro 5.2); unverified.

AMS 30 · solved (literature)

theorem erdos_119.parts.i :
    answer(True) ↔ ∀ (z : ℕ → ℂ) (hz : ∀ i : ℕ, ‖z i‖ = 1),
      atTop.limsup (fun n => (M z n : EReal)) = ⊤

open