Asymptotics of the spectrum of partial theta function

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 partial theta function. For any fixed $q\in (0,1)$ it has infinitely many negative zeros. For $q$ taking one of the 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 $q\ne \tilde{q}_j$ the partial theta function has no multiple real zeros. We prove that $\tilde{q}_j=1-(\pi /2j)+o(1/j)$ and that $\lim _{j\rightarrow \infty }y_j=-e^{\pi }=-23.1407\ldots$.