On the double zeros of a partial theta function

TL;DR

This paper investigates the behavior of zeros of the partial theta function, especially the occurrence of double zeros at spectral values of q, providing asymptotic formulas for these zeros and their corresponding spectral points.

Contribution

The paper characterizes the spectral values of q where the partial theta function has double zeros and derives their asymptotic behavior, a novel analysis of zero multiplicities in this context.

Findings

Spectral q-values approach 1 as j increases.

Double zeros occur only at spectral q-values.

Asymptotic formulas for spectral q-values and zeros.

Abstract

The series $\theta (q,x):=\sum _{j=0}^{\infty}q^{j(j+1)/2}x^j$ converges for $q\in [0,1)$, $x\in \mathbb{R}$, and defines a {\em partial theta function}. For any fixed $q\in (0,1)$ it has infinitely many negative zeros. For $q$ taking one of the {\em spectral} values $\tilde{q}_1$, $\tilde{q}_2$, $\ldots$ (where $0.3092493386\ldots =\tilde{q}_1<\tilde{q}_2<\cdots <1$, $\lim _{j\rightarrow \infty}\tilde{q}_j=1$) the function $\theta (q,.)$ has a double zero $y_j$ which is the rightmost of its real zeros (the rest of them being simple). For $q\neq \tilde{q}_j$ the partial theta function has no multiple real zeros. We prove that $\tilde{q}_j=1-\pi /2j+(\log j)/8j^2+O(1/j^2)$ and $y_j=-e^{\pi}e^{-(\log j)/4j+O(1/j)}$.