On the multiple zeros of a partial theta function

TL;DR

This paper investigates the zeros of the partial theta function, establishing an upper bound on the magnitude of its multiple zeros for fixed parameters.

Contribution

It provides a bound on the size of multiple zeros of the partial theta function for fixed q, a novel result in the analysis of this special function.

Findings

Multiple zeros are bounded by 8^{11} in magnitude.

The result applies for any fixed q with 0<|q|<1.

The paper advances understanding of the zero structure of the partial theta function.

Abstract

We consider the partial theta function $\theta (q,x):=\sum_{j=0}^{\infty}q^{j(j+1)/2}x^j$, where $x\in \mathbb{C}$ is a variable and $q\in \mathbb{C}$, $0<|q|<1$, is a parameter. We show that, for any fixed $q$, if $\zeta$ is a multiple zero of the function $\theta (q,.)$, then $|\zeta |\leq 8^{11}$.