# Holomorphic functions on the disk with infinitely many zeros

### Sebastiano Boscardin

Scuola Normale Superiore, Pisa, Italy

## Abstract

Suppose that $ϕ(z)=∑_{n=1}d_{n}z_{n}$ is a series with radius of convergence greater than 1 and suppose that $d_{n}$ are real numbers such that $ϕ(1)=0$. We prove that for any integer $a$ greater than $1$, the complex variable function

$f(z)=n=0∑∞ ϕ(z_{a_{n}})=ϕ(z)+ϕ(z_{a})+ϕ(z_{a_{2}})+ϕ(z_{a_{3}})+⋯$

has infinitely many zeros on the unit disk. It even takes every complex value in every disk centered in any point of the boundary.

## Cite this article

Sebastiano Boscardin, Holomorphic functions on the disk with infinitely many zeros. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. 34 (2023), no. 2, pp. 465–490

DOI 10.4171/RLM/1015