Two properties of the partial theta function

TL;DR

This paper investigates the zero set of the partial theta function, proving its connectedness and smoothness at certain zeros, and analyzes the asymptotic distribution of real zeros as the parameter approaches specific boundary values.

Contribution

It establishes the connectedness and smoothness properties of the zero set of the partial theta function and describes the asymptotic behavior of its real zeros near boundary points.

Findings

Zero set of the partial theta function is connected.

Zero set is smooth at simple or double zeros.

Asymptotic distribution of real zeros near boundary values.

Abstract

For the partial theta function $\theta (q,z):=\sum_{j=0}^{\infty}q^{j(j+1)/2}z^j$, $q$, $z\in \mathbb{C}$, $|q|<1$, we prove that its zero set is connected. This set is smooth at every point $(q^{\flat},z^{\flat})$ such that $z^{\flat}$ is a simple or double zero of $\theta (q^{\flat},.)$. For $q\in (0,1)$, $q\rightarrow 1^-$ and $a\geq e^{\pi}$, there are $o(1/(1-q))$ and $(\ln (a/e^{\pi}))/(1-q)+o(1/(1-q))$ real zeros of $\theta (q,.)$ in the intervals $[-e^{\pi},0)$ and $[-a,-e^{-\pi}]$ respectively (and none in $[0,\infty)$). For $q\in (-1,0)$, $q\rightarrow -1^+$ and $a\geq e^{\pi /2}$, there are $o(1/(1+q))$ real zeros of $\theta (q,.)$ in the interval $[-e^{\pi /2},e^{\pi /2}]$ and $(\ln (a/e^{\pi /2})/2)/(1+q)+o(1/(1+q))$ in each of the intervals $[-a,-e^{\pi /2}]$ and $[e^{\pi /2},a]$.