Zeros of Sections of Power Series: Deterministic and Random
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Zeros of Sections of Power Series: Deterministic and Random José L. Fernández1
Received: 14 June 2016 / Revised: 23 November 2016 / Accepted: 2 December 2016 © Springer-Verlag Berlin Heidelberg 2017
Abstract We present a streamlined proof (and some refinements) of a characterization (due to F. Carlson and G. Bourion, and also to P. Erd˝os and H. Fried) of the socalled Szeg˝o power series. This characterization is then applied to readily obtain some (more) recent known results and some new results on the asymptotic distribution of zeros of sections of random power series, extricating quite naturally the deterministic ingredients. Finally, we study the possible limits of the zero counting probabilities of a power series. Keywords Power series · Random power series · Zeros · Sections · Jentzsch–Szeg˝o theorem · Equidistribution Mathematics Subject Classification 30B20 · 30B10
1 Introduction The first aim of this paper is to present a streamlined proof and a refined version of a characterization (due to F. Carlson and G. Bourion, and also to P. Erd˝os and H. Fried) of the so-called Szeg˝o power series: Theorems 2.6 and 2.7.
To Juha Heinonen, in memoriam. Communicated by Dmitry Khavinson. Research partially supported by Fundación Akusmatika.
B 1
José L. Fernández [email protected] Departamento de Matemáticas, Universidad Autónoma de Madrid, 28049 Madrid, Spain
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J. L. Fernández
That characterization is then applied in Sect. 5 to readily obtain some (more) recent known results and some new results on the asymptotic distribution of zeros of sections of random power series, extricating quite naturally the deterministic ingredients. Finally, in Sect. 6 we study the possible limits of the zero counting probabilities associated to a power series. We shall denote by F the class of power series whose radius of convergence is 1. The results which we are about to discuss concerning such f can be translated, with obvious scaling, to power series of positive and finite radius of convergence. For a given power series f ∈ F and for each n ≥ 0, we denote by sn = sn ( f ) the nth section of the power series: sn (z) = nk=0 ak z k , and by Zn the (multi-)set of the zeros of sn . To each non-constant sn we associate two measures: we denote by μn = μn ( f ) the zero counting measure μn =
1 δw , n w∈Zn
a weighted sum of Dirac deltas placed at the zeros of sn repeated according to their multiplicity, and we denote by ρn = ρn ( f ) the circular projection of μn : ρn =
1 δ|w| . n w∈Zn
If an = 0, then μn and ρn are probability measures. If an = 0 (and sn is non-constant), we append the definition above by adding a Dirac delta at ∞C with mass n − deg(sn ) C. By Fn we so that μn and ρn become probability measures on the Riemann sphere denote the distribution function of ρn , given by Fn (t) = μn (|z| ≤ t), for t ≥ 0, thus Fn (t) is the average number of zeros of sn within the disk {z ∈ C : |z| ≤ t}. By Hurwitz’s theorem, limn→∞ Fn (t) = 0 for any t < 1; actually, Fn (t) = O(1/n), for any fixed t < 1. We are concer
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