Some analytic properties of the partial theta function

We prove new properties of the zero set of Ramanujan’s partial theta function $\theta (q,x):={\sum }_{j=0}^{\mathrm{\infty }}{q}^{j(j+1)\mathrm{/}2}{x}^j$, $q\in (-1,0)\cup (0,1)$, $x\in \mathbb{R}$. We show that for each $q\in (0,1)$, there exists a line Re$x=-a$, $a\ge 5$, such that all real zeros of $\theta (q,\mathrm{.})$ lie to its left and all complex zeros to its right. A similar property is proved for $q\in (-1,0)$. For $q\in (0,1)$, there are no real zeros $\ge -6$. For $q\in (-1,0)$, there are no negative zeros $\ge -2.4$ and no positive zeros $\le 2.4$, except the smallest one.