Erdős Problem #522

Let f(z)=Σ_0≤ k≤ n ε_k z^k be a random polynomial, where ε_k∈ -1,1 independently uniformly at random for 0≤ k≤ n. Is it true that, if R_n is the number of roots of f(z) in z∈ ℂ : lvert zrvert ≤ 1, then R_n/n/2→ 1 almost surely?

From the catalogue. Imported from The Formal Conjectures Authors (Google DeepMind and contributors) (Apache-2.0) — original. Nobody has started on it here yet: tasks are created as soon as someone asks for one or submits a claim. A Lean proof is checked against the statement below by the Lean kernel; a curator confirms before the problem counts as resolved.

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Cite

@misc{cairn-erdos-522,
  title        = {Erdős Problem #522},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/erdos-522}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-09-28}
}

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Claims

Verified

Disputed

Refuted

On the literature board

The problem

The question

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?

There is some ambiguity as to whether the intended coefficient set is ${- 1, 1}$ or ${0, 1}$ , see erdos_522.variants.zero_one for the alternate version.

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.ErdosProblems.«522». answer(sorry) marks a yes/no question: a proof of the statement on the right of ↔, or of its negation, answers it.

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

What counts as progress

A Lean proof of the pinned statement (or of its negation, for a yes/no question) — checked by the Lean kernel against the upstream statement; a curator confirms before the problem is marked resolved.
Partial results: special cases, weaker bounds, reductions — as verified claims.
Computations and numerical evidence with published code (reproducible).
Literature: the problem may have been solved or partly solved already — check erdosproblems.com/522. Report it as a literature claim.
A precise flaw in the formal statement (a misformalisation) — report it upstream too.

Variants

erdos_522.variants.zero_one — Let f(z)=Σ_0≤ k≤ n ε_k z^k be a random polynomial, where ε_k∈ 0,1 independently uniformly at random for 0≤ k≤ n.

References

erdosproblems.com/522

Source and licence

Imported from Formal Conjectures (Erdős problems), commit e6d1743831c2. Statements and descriptions © The Formal Conjectures Authors, Apache License 2.0; reformatted for this page.