On the Behavior of Power Series with Completely Additive Coefficients

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.