On the location of the complex conjugate zeros of the partial theta function

Abstract: We prove that for any $q\in (0,1)$, all complex conjugate pairs of zeros of the partial theta function $\theta (q,x):=\sum _{j=0}^{\infty}q^{j(j+1)/2}x^j$ with non-negative real part belong to the half-annulus ${$Re$(x)\geq 0,~1<|x|<5}$, where the outer radius cannot be replaced by a number smaller than $e^{\pi /2}=4.810477382\ldots$, and that for $q\in (0,0.2^{1/4}=0.6687403050\ldots ]$, $\theta (q,.)$ has no zeros with non-negative real part. The complex conjugate pairs of zeros with negative real part belong to the left open half-disk of radius $49.8$ centered at the origin.