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 \(\lim _{j\rightarrow \infty }y_j=-e^{\pi }=-23.1407\ldots \).