Let f(z)=∑_0≤ k≤ n epsilon_k z^k be a random polynomial, where epsilon_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 : | z| ≤ 1, then fracR_nn/2→ 1 almost surely?

Erdős Problem #522 — Let f(z)=∑_0≤ k≤ n epsilon_k z^k be a random polynomial, where epsilon_k∈ -1,1 independently uniformly at random for 0≤ k≤ n.

Erdős Problem #522 asks whether the proportion of roots inside the unit disk approaches one-half almost surely for random sign polynomials. The coefficient convention may be ambiguous between $\{-1,1\}$ and $\{0,1\}$ ; the $\{-1,1\}$ version is the one treated as open.

Known results

Erdős and Offord (1956): the number of real roots is $\left(\frac{2}{\pi}+o(1)\right)\log n$ .
Yakir (2021): $R_n=n/2+O(n^{9/10})$ in probability, proving $R_n/(n/2)\to1$ in probability.

Quantitative almost-sure claim

A repository claims that $R_n= n/2+O(n^{7/8+\delta})$ almost surely for every $\delta>0$ , which would solve the problem. This remains unverified: no retrieved preprint or peer-reviewed source corroborates it, and the formalization still uses sorry.

Current status (as of March 2026): the in-probability result is established, while the stronger almost-sure convergence has only an unverified claimed proof.

Erdős Problem #522 — Let $f(z)=\sum_{0\leq k\leq n} \epsilon_k z^k$ be a random polynomial, where $\epsilon_k\in \{-1,1\}$ independently uniformly at random for $0\leq k\leq n$ .

Primary source

Additional references

Progress summary

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