Approximately half of the roots of a random Littlewood polynomial are inside the disk
Volume 261 / 2021
Studia Mathematica 261 (2021), 227-240
MSC: 30C15, 26C10, 60G55.
DOI: 10.4064/sm201117-28-1
Published online: 31 May 2021
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 1967 book.
