On the multiple zeros of a partial theta function

Abstract: We consider the partial theta function $\theta (q,x):=\sum_{j=0}^{\infty}q^{j(j+1)/2}x^j$, where $x\in \mathbb{C}$ is a variable and $q\in \mathbb{C}$, $0<|q|<1$, is a parameter. We show that, for any fixed $q$, if $\zeta$ is a multiple zero of the function $\theta (q,.)$, then $|\zeta |\leq 8^{11}$.