On the Behavior of Power Series with Completely Additive Coefficients

Oleg Petrushov¹

¹ Moscow State University Moscow, Russia

Bulletin of the Polish Academy of Sciences. Mathematics, Tome 63 (2015) no. 3, pp. 217-225

Résumé

Consider the power series $\mathfrak {A}(z)= \sum _{n=1}^{\infty }\alpha (n)z^n$, where $\alpha (n)$ is a completely additive function satisfying the condition $\alpha (p)=o(\operatorname {ln}p)$ for prime numbers $p$. Denote by $e(l/q)$ the root of unity $e^{2\pi il/q}$. We give effective omega-estimates for $\mathfrak {A}(e(l/p^k)r)$ when $r\to 1-$. From them we deduce that if such a series has non-singular points on the unit circle, then it is a zero function.

Détail

DOI : 10.4064/ba8018-1-2016

Keywords: consider power series mathfrak sum infty alpha where alpha completely additive function satisfying condition alpha operatorname prime numbers denote root unity effective omega estimates mathfrak k deduce series has non singular points unit circle zero function