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

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).

arXiv comment14 pages