On the zero set of the partial theta function

TL;DR

This paper analyzes the zero set of the partial theta function for real and complex arguments, revealing countably many smooth curves and the behavior of zeros, including double zeros and their crossing properties.

Contribution

It provides a detailed description of the zero set structure of the partial theta function for q in (-1,0) and (0,1), including the nature of double zeros and zero crossings.

Findings

Zero set consists of countably many smooth curves in the (q,x)-plane.

Double zeros occur at specific x-intervals depending on q.

Complex conjugate zeros cross the imaginary axis for q in (0,1).

Abstract

We consider the partial theta function $\theta (q,x):=\sum _{j=0}^{\infty}q^{j(j+1)/2}x^j$, where $q\in (-1,0)\cup (0,1)$ and either $x\in \mathbb{R}$ or $x\in \mathbb{C}$. We prove that for $x\in \mathbb{R}$, in each of the two cases $q\in (-1,0)$ and $q\in (0,1)$, its zero set consists of countably-many smooth curves in the $(q,x)$-plane each of which (with the exception of one curve for $q\in (-1,0)$) has a single point with a tangent line parallel to the $x$-axis. These points define double zeros of the function $\theta (q,.)$; their $x$-coordinates belong to the interval $[-38.83\ldots ,-e^{1.4}=4.05\ldots )$ for $q\in (0,1)$ and to the interval $(-13.29,23.65)$ for $q\in (-1,0)$. For $q\in (0,1)$, infinitely-many of the complex conjugate pairs of zeros to which the double zeros give rise cross the imaginary axis and then remain in the half-disk $\{ |x|<18$, Re\,$x>0\}$. For $q\in (-1,0)$, complex conjugate pairs do not cross the imaginary axis.