Addendum: A separation in modulus property of the zeros of a partial theta function

Abstract

We consider the partial theta function $\theta (q,z):=\sum _{j=0}^{\infty}q^{j(j+1)/2}z^j$, where $z\in \mathbb{C}$ is a variable and $q\in \mathbb{C}$, $0<|q|<1$, is a parameter. Set $D(a):=\{ q\in \mathbb{C}$, $0<|q|\leq a$, $\arg (q)\in [\pi /2,3\pi /2]\}$. We show that for $k\in \mathbb{N}$ and $q\in D(0.55)$, there exists exactly one zero of $\theta (q,.)$ (which is a simple one) in the open annulus $|q|^{-k+1/2}<z<|q|^{-k-1/2}$ (if $k\geq 2$) or in the punctured disk $0<z<|q|^{-3/2}$ (if $k=1$). For $k=1$, $4$, $5$, $6$, $\ldots$, this holds true for $q\in D(0.6)$ as well.