# Zeros of Normalized Sections of Non Convergent Power Series

## TL;DR

This paper extends Carlson's classical result on the distribution of zeros of power series sections to include those with zero radius of convergence, using appropriate normalizations.

## Contribution

It generalizes the known zero distribution characterization from convergent to non-convergent power series with necessary normalizations.

## Key findings

- Zeros of normalized sections are asymptotically equidistributed on |z|=R for non-convergent series
- Extension of Carlson's result to series with null radius of convergence
- Provides conditions under which zero distribution behavior is characterized

## Abstract

A well known result due to Carlson affirms that a power series with finite and positive radius of convergence R has no Ostrowski gaps if and only if the sequence of zeros of its nth sections is asymptotically equidistributed to {|z|=R}. Here we extend this characterization to those power series with null radius of convergence, modulo some necessary normalizations of the sequence of the sections of f.