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}$$isavariableand$q\in $\mathbb{C}$$,$0<|q|<1$,isaparameter\mathrm{.}Set$\alpha _0~:=~\sqrt{3}/2\pi ~=~0.2756644477\ldots$\mathrm{.}Weshowthat,for$n\geq 5$,for$|q|\leq 1-1/(\alpha _0n)$andfor$k\geq n$thereexistsauniquezero$\xi _k$of$\theta (q,.)$satisfyingtheinequalities$|q|^{-k+1/2}<|\xi _k|<|q|^{-k-1/2}$; all these zeros are simple ones. The moduli of the remaining $n-1$ zeros are $\leq |q|^{-n+1/2}$\mathrm{.}A\text{em}spectralvalueof$q$isavalueforwhich$\theta (q,.)$hasamultiplezero\mathrm{.}Weprovetheexistenceofthespectralvalues$0.4353184958\ldots \pm i, 0.1230440086\ldots$forwhich$\theta$ has double zeros $-5.963\dots \pm i6.104\dots$.