Erdős Problem #1039

Erdős problem #1039 · erdosproblems.com/1039

For $f(z) = \prod_{i=1}^n (z - z_i)$ with all $|z_i| \le 1$, let $\rho(f)$ be the radius of the largest disc contained in $\{z : |f(z)| < 1\}$. Is $\rho(f) \gg 1/n$? The worst case is now known to be $\Theta(1/n)$, with the explicit bound $\rho(f) \ge (\log 2)/n$.

Result

Proved(see note)

Status

Resolved

AI contribution

AI co-developed

Method

Argument

Field

Complex Analysis

Posed by

—

Year posed

1958

Years open

68y

Solved

2026-05-17

Model

GPT-5.5 Pro, Codex 5.5

Vendor

OpenAI

Collaborators

—

Verification

Lean-verified

Publication

Announced

Significance

10 / 100*

Disclosed cost

—

Wikipedia

No dedicated article