Means of algebraic numbers in the unit disk

TL;DR

This paper investigates the behavior of the arithmetic means of zeros of polynomials with integer coefficients in the unit disk, proving they tend to zero and providing convergence rate estimates.

Contribution

It extends Schur's classical results by proving the means tend to zero and generalizes the Erdős-Turán theorem to estimate convergence rates.

Findings

Arithmetic means of zeros tend to zero.

Established convergence rate estimates.

Generalized Erdős-Turán theorem for polynomial zeros.

Abstract

Schur studied limits of the arithmetic means $s_n$ of zeros for polynomials of degree $n$ with integer coefficients and simple zeros in the closed unit disk. If the leading coefficients are bounded, Schur proved that $\limsup_{n\to\infty} |s_n| \le 1-\sqrt{e}/2.$ We show that $s_n \to 0$, and estimate the rate of convergence by generalizing the Erd\H{o}s-Tur\'an theorem on the distribution of zeros.