A domain free of the zeros of the partial theta function

The partial theta function is the sum of series \medskip\centerline{$\displaystyle\theta (q,x):=\sum\nolimits _{j=0}^{\infty}q^{j(j+1)/2}x^j$,}\medskip\noi where $q$is a real or complex parameter ($|q|<1$). Its name due to similaritieswith formula for Jacobi function$\Theta (q,x):=\sum _{j=-\infty}^{\infty}q^{j^2}x^j$. $\theta$ has been considered in Ramanujan's lost notebook. Itfinds applicationsin several domains, such as Ramanujan type$q$-series, theory (mock) modular forms, asymptotic analysis, statistical physics, combinatorics and most recently study section-hyperbolic polynomials,i.~e. polynomials with all coefficients positive,with roots negative whose sections (i.~e. truncations)are also real-rooted.For each $q$ fixed,$\theta$ an entire order $0$ variable~$x$. When$q$ $q\in (0,0.3092\ldots )$, $\theta (q,.)$ theLaguerre-P\'olyaclass $\mathcal{L-P}I$. More generally, $q \in (0,1)$, product realpolynomialwithout zeros class Thus it isan withinfinitely-many negative, no positive finitely-many complexconjugate zeros. latter are known belongto explicitly defined compact domain containing andindependent while tend infinity ageometric progression ratio $1/q$. A similar result true for$q\in (-1,0)$ when there infinitely-many zeros.We consider thequestion how close origin can be.In generalcase truethat their moduli always larger than $1/2|q|$. We case prove that any (0,1)$,the on set $$\displaystyle \{x\in\mathbb{C}\colon |x|\leq 3\} \cap {\rm Re} x\leq 0\}\cap |{\rm Im} x|\leq 3/\sqrt{2}\}$$which containsthe closure left unit half-disk more $7$ times it.It unlikely this hold whole lefthalf-disk radius~$3$. Similar domains do not exist Re$x\geq 0$, (-1,0)$, 0$ Re$x\leq 0$. show alsothat $\geq -5$ (but one find $-7.51$).