Abstract
The classical theorems of Jentzsch and Szegö concern the limiting behavior of the zeros of the partial sums of a power series. More precisely, if
are the partial sums of a power series EquationSource% MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzamaabm % aabaGaamOEaaGaayjkaiaawMcaaiabg2da9maaqaeabaWaa0baaSqa % aiaadUgacqGH9aqpcaaIWaaabaGaeyOhIukaaOGaamyyamaaBaaale % aacaWGRbaabeaakiaadQhadaahaaWcbeqaaiaadUgaaaaabeqab0Ga % eyyeIuoaaaa!44F6!]]</EquationSource><EquationSource Format="TEX"><![CDATA[$$f\left( z \right) = \sum {_{k = 0}^\infty {a_k}{z^k}} $$ having finite positive radius of convergence ρ, then Jentzsch [91] proved that each point of the circle of convergence C ρ := {z: |z| = ρ} is a limit point of zeros of polynomials s n (z), n = 1, 2,... . Szegö [170] substantially improved this result by showing that there is a subsequence EquationSource% MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaiWaaeaaca % WGUbWaaSbaaSqaaiaadUgaaeqaaaGccaGL7bGaayzFaaWaa0baaSqa % aiaadUgacqGH9aqpcaaIXaaabaGaeyOhIukaaaaa!3E8D!]]</EquationSource><EquationSource Format="TEX"><![CDATA[$$\left\{ {{n_k}} \right\}_{k = 1}^\infty $$ for which the zeros of the partial sums EquationSource% MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4CamaaBa % aaleaacaWGUbaabeaakmaaBaaaleaadaWgaaadbaGaam4Aaaqabaaa % leqaaOWaaeWaaeaacaWG6baacaGLOaGaayzkaaaaaa!3BFB!]]</EquationSource><EquationSource Format="TEX"><![CDATA[$${s_n}_{_k}\left( z \right)$$ are uniformly distributed in angle; that is, if S(α, β) is the sector
, and Z n (A) denotes the number of zeros of s n in the set A, then
for all sectors S(α, β).
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Historical Comments
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Andrievskii, V.V., Blatt, HP. (2002). Zero Distribution of Polynomials. In: Discrepancy of Signed Measures and Polynomial Approximation. Springer Monographs in Mathematics. Springer, New York, NY. https://doi.org/10.1007/978-1-4757-4999-1_2
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