A domain containing all zeros of the partial theta function

TL;DR

This paper characterizes the location of all zeros of the partial theta function, showing they are confined to specific regions in the complex plane depending on the parameter q.

Contribution

It provides a detailed description of the zero distribution of the partial theta function for all q in (0,1), a novel geometric insight.

Findings

Zeros are confined to Re z<0 with bounded Im z

Zeros with Re z≥0 lie within a circle of radius 18

Zero regions depend explicitly on the parameter q

Abstract

We consider the partial theta function, i.e. the sum of the bivariate series $\theta (q,z):=\sum_{j=0}^{\infty}q^{j(j+1)/2}z^j$ for $q\in (0,1)$, $z\in \mathbb{C}$. We show that for any value of the parameter $q\in (0,1)$ all zeros of the function $\theta (q,.)$ belong to the domain $\{ {\rm Re}~z<0, |{\rm Im}~z|\leq 132\}$$\cup$$\{ {\rm Re}~z\geq 0, |z|\leq 18\}$.