Addendum: 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$D(a):={ q\in $\mathbb{C}$$,$0<|q|\leq a$,$\arg (q)\in [\pi /2,3\pi /2]}$\mathrm{.}Weshowthatfor$k\in $\mathbb{N}$$and$q\in D(0.55)$,thereexistsexactlyonezeroof$\theta (q,.)$(whichisasimpleone)intheopenannulus$|q|^{-k+1/2}<z<|q|^{-k-1/2}$(if$k\geq 2$)orinthepunctureddisk$0<z<|q|^{-3/2}$(if$k=1$)\mathrm{.}For$k=1$,$4$,$5$,$6$,$\ldots$,thisholdstruefor$q\in D(0.6)$ as well.