Problem Statement
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 $0\leq k\leq n$.
Does there exist some constant $C>0$ such that, almost surely,\[\max_{\lvert z\rvert=1}\left\lvert \sum_{k\leq n}\epsilon_k(t)z^k\right\rvert=(C+o(1))\sqrt{n\log n}?\]
Does there exist some constant $C>0$ such that, almost surely,\[\max_{\lvert z\rvert=1}\left\lvert \sum_{k\leq n}\epsilon_k(t)z^k\right\rvert=(C+o(1))\sqrt{n\log n}?\]
Categories:
Analysis Probability Polynomials
Progress
Salem and Zygmund [SZ54] proved that $\sqrt{n\log n}$ is the right order of magnitude, but not an asymptotic.This was settled by Halász [Ha73], who proved this is true with $C=1$.
Source: erdosproblems.com/523 | Last verified: January 15, 2026