El teorema d’Erdős-Turán

Abstract

[en] A classical result of Erdős and Turán states that if a monic polynomial has small size on the unit circle and its constant coefficient is not too small, then its zeros cluster near the unit circle and become equidistributed in angle. The theorem of Erdős and Turán are then two results: that the zeros of a polynomial lie close to the unit circle and that the angles of the zeros are well distributed. The first result (Theorem 1 p.4) is a simple consequence of Jensen’s formula. The second (Theorem 2 p.5), which is the main result of the paper, we will prove by seeing that the discrepancy of the angles of the zeros of a polynomial is bounded by a measure of the size of the polynomial at the circle. To prove these results we will follow the article by K. Soundararajan [14].