Comptes Rendus
Number Theory
Means of algebraic numbers in the unit disk
Comptes Rendus. Mathématique, Volume 347 (2009) no. 3-4, pp. 119-122.

Schur studied limits of the arithmetic means sn 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 supn|sn|1e/2. We show that sn0, and estimate the rate of convergence by generalizing the Erdős–Turán theorem on the distribution of zeros.

Schur a étudié les limites des moyennes arithmétiques sn des zéros pour les polynômes à coefficients entiers de degré n ayant des zéros simples dans le disque unité fermé. Lorsque les coefficients dominants restent bornés, Schur a démontré que lim supn|sn|1e/2. Nous prouvons que sn0. Nous donnons une estimation du taux de convergence, grâce à une généralisation d'un théorème de Erdős–Turán sur la distribution des zéros.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2009.01.002

Igor E. Pritsker 1

1 Department of Mathematics, Oklahoma State University, Stillwater, OK 74078, USA
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Igor E. Pritsker. Means of algebraic numbers in the unit disk. Comptes Rendus. Mathématique, Volume 347 (2009) no. 3-4, pp. 119-122. doi : 10.1016/j.crma.2009.01.002. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2009.01.002/

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