Approximately half of the roots of a random Littlewood polynomial are inside the disk

TL;DR

This paper proves that for large degrees, nearly half of the roots of a random Littlewood polynomial lie inside the unit disk, resolving a longstanding problem in function theory.

Contribution

It establishes that almost all Littlewood polynomials of large degree have about half their roots inside the unit disk, solving a problem posed in 1967.

Findings

Approximately half of the roots are inside the unit disk for large n

The number of exceptions is negligible compared to 2^n

The result confirms a conjecture from Hayman's research problems

Abstract

We prove that for large $n$, all but $o(2^{n})$ polynomials of the form $P(z) = \sum_{k=0}^{n-1}\pm z^k$ have $n/2 + o(n)$ roots inside the unit disk. This solves a problem from Hayman's book 'Research Problems in Function Theory' (1967).