Erdős Problem 522: Zero One

Let $f(z)=\sum_{0\leq k\leq n} \epsilon_k z^k$ be a random polynomial, where $\epsilon_k\in \{0,1\}$ independently uniformly at random for $0\leq k\leq n$ .

Field theory and polynomials AMS 12 AMS 60

Mathematical statement

Let $f(z)=\sum_{0\leq k\leq n} \epsilon_k z^k$ be a random polynomial, where $\epsilon_k\in \{0,1\}$ 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 $\{ z\in \mathbb{C} : \lvert z\rvert \leq 1\}$ , then

\frac{R_n}{n/2}\to 1

almost surely?

Statement source: Erdős Problems statement material

Statement terms: Source-specific

Source-specific terms. Therefore does not assert reuse rights beyond attributed display.

Statement artifacts, not proofs

These records expose exact Lean propositions and statement-only wrappers. Defining a proposition does not supply a proof of it. A placeholder-bearing target also contains no proof. Elaboration checks syntax and types; it does not certify that a formalization perfectly captures every nuance of the informal problem.

Pinned Lean formulation 1

erdos_522.variants.zero_one

Canonical source

Complete statement target, proof intentionally absentLean 4

theorem erdos_522.variants.zero_one :    answer(sorry) ↔ ∀ {Ω : Type*} [MeasureSpace Ω] [IsProbabilityMeasure (ℙ : Measure Ω)]      {n : ℕ} (hn : 1 ≤ n) (f : KacCoefficients ({0, 1} : Set ℂ) Ω),      ℙ {ω | atTop.Tendsto (fun n : ℕ ↦ (2 * f.numRootsInUnitDisk n ω : ℝ) / n) (𝓝 1)} = 1 := by  sorry

Statement source: Formal Conjectures
Lean version: v4.27.0
Placeholder: Present; no proof artifact
Source evidence: Pinned source index
Fidelity review: Community formulation

References

Pinned Formal Conjectures sourceformalization erdosproblems.com/522background