Holomorphic functions on the disk with infinitely many zeros

Abstract

Suppose that $\phi(z)=\sum\_{n=1}^\infty d\_nz^n$ is a series with radius of convergence greater than 1 and suppose that $d\_n$ are real numbers such that $\phi(1)\neq0$. We prove that for any integer $a$ greater than $1$, the complex variable function $$ f(z)=\sum\_{n=0}^{\infty}\phi(z^{a^n})=\phi(z)+\phi(z^a)+\phi(z^{a^2})+\phi(z^{a^3})+\cdots $$ has infinitely many zeros on the unit disk. It even takes every complex value in every disk centered in any point of the boundary.