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

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 $\alpha _0~:=~\sqrt{3}/2\pi ~=~0.2756644477\ldots$. We show that, for $n\geq 5$, for $|q|\leq 1-1/(\alpha _0n)$ and for $k\geq n$ there exists a unique zero $\xi _k$ of $\theta (q,.)$ satisfying the inequalities $|q|^{-k+1/2}<|\xi _k|<|q|^{-k-1/2}$; all these zeros are simple ones. The moduli of the remaining $n-1$ zeros are $\leq |q|^{-n+1/2}$. A {\em spectral value} of $q$ is a value for which $\theta (q,.)$ has a multiple zero. We prove the existence of the spectral values $0.4353184958\ldots \pm i\, 0.1230440086\ldots$ for which $\theta$ has double zeros $-5.963\ldots \pm i\, 6.104\ldots$.