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

Oleg A. Petrushov¹

¹ Moscow State University Vorobyovi Gory, Russia

Acta Arithmetica, Tome 180 (2017) no. 4, pp. 319-332.

Résumé

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.

Détail

DOI : 10.4064/aa8536-4-2017

Keywords: consider power series mathfrak sum infty alpha where alpha additive function satisfying condition alpha where infty uniformly respect nbsp denote root unity series effective omega estimates mathfrak k estimates deduce series has non singular points unit circle rational function