Means of algebraic numbers in the unit disk

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 lim sup n → ∞ | s n | ⩽ 1 − e / 2 . We show that s n → 0 , and estimate the rate of convergence by generalizing the Erdős–Turán theorem on the distribution of zeros. To cite this article: I.E. Pritsker, C. R. Acad. Sci. Paris, Ser. I 347 (2009).