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$(x)\ge 0,1<|x|<5\}$, where the outer radius cannot be replaced by a number smaller than $e^{\pi /2}=4.810477382\dots$, and that for $q\in (0,0.2^{1/4}=0.6687403050\dots ]$, $\theta (q,.)$ 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.

18 pages, 1 figure