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On the behavior close to the unit circle of power series with additive coefficients

Volume 180 / 2017

Oleg A. Petrushov Acta Arithmetica 180 (2017), 319-332 MSC: Primary 11N37; Secondary 30B30. DOI: 10.4064/aa8536-4-2017 Published online: 1 September 2017

Abstract

Consider the power series $\mathfrak{A}(z)= \sum_{n=1}^{\infty}\alpha(n)z^n$, where $\alpha(n)$ is an additive function satisfying the condition $\alpha(p^m)=mf(p,m)\ln p$, where $f(p,m)\to 0$ as $p\to \infty$ uniformly with respect to $m$. Denote by $e(l/q)$ the root of unity $e^{2\pi il/q}$. For such series we give effective omega-estimates for $\mathfrak{A}(e(l/p^k)r)$ as $r\to 1-$. From the estimates we deduce that if such a series has non-singular points on the unit circle then it is a rational function.

Authors

  • Oleg A. PetrushovMoscow State University
    Vorobyovi Gory, Russia
    e-mail

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