On the spectrum of a partial theta function

TL;DR

This paper investigates the zero distribution of the partial theta function, establishing that for small enough |q|, specifically up to 0.108, the function has no multiple zeros, contributing to the understanding of its spectral properties.

Contribution

The paper proves that the partial theta function has no multiple zeros for all |q| ≤ 0.108, providing new insights into its zero structure.

Findings

No multiple zeros for |q| ≤ 0.108

Spectral properties of the partial theta function clarified

Zero distribution analyzed for small |q|

Abstract

The bivariate series $\theta (q,x):=\sum_{j=0}^{\infty}q^{j(j+1)/2}x^j$ defines a {\em partial theta function}. For fixed $q$, $\theta (q,.)$ is an entire function. We show that for $|q|\leq 0.108$ the function $\theta (q,.)$ has no multiple zeros.