No zeros of the partial theta function in the unit disk

TL;DR

This paper proves that the partial theta function has no zeros within the closed unit disk for all real q in the interval (-1,1), extending understanding of its zero distribution.

Contribution

It establishes a zero-free region for the partial theta function in the unit disk for all real q in (-1,1), a novel result in the study of special functions.

Findings

No zeros of the partial theta function in the unit disk for q in (-1,1)

Zero-free region extends to all real q in the specified interval

Enhances understanding of the partial theta function's analytic properties

Abstract

We prove that for $q\in (-1,0)\cup (0,1)$, the partial theta function $\theta (q,x):=\sum _{j=0}^{\infty}q^{j(j+1)/2}x^j$ has no zeros in the closed unit disk.