Let $f (z) = \sum_{0 \leq k \leq n} ϵ_{k} z^{k}$ be a random polynomial, where $ϵ_{k} \in {- 1, 1}$ independently uniformly at random for $0 \leq k \leq n$ . Is it true that, if $R_{n}$ is the number of roots of $f (z)$ in ${z \in C : ∣ z ∣ \leq 1}$ , then $\frac{R _{n}}{n /2} \to 1$ almost surely?

unreviewedOpen. Worked here; no verified result yet.

analysis · open · formalized (Lean) · 0 attempts

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Evidence

unverified AI candidates (2)

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

Let (f_n(z)=\sum_{k=0}^n \epsilon_k z^k) with i.i.d. (\epsilon_k\in{-1,1}), and let (R_n) be the number of zeros in ({|z|\le 1}).

candidate solution ↗

llm-hunter · gpt pro 5.2 · unverified

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

candidate solution ↗

Formal proof

AMS 12 60 · open (literature)

theorem erdos_522 :
    answer(sorry) ↔ ∀ {Ω : Type*} [MeasureSpace Ω] [IsProbabilityMeasure (ℙ : Measure Ω)]
      (c : KacCoefficients ({-1, 1} : Set ℂ) Ω),
      ℙ {ω | atTop.Tendsto (fun n : ℕ ↦ (2 * c.numRootsInUnitDisk n ω : ℝ) / n) (𝓝 1)} = 1

formal-conjectures/522.lean ↗

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