Some analytic properties of the partial theta function

Abstract

We prove new properties of the zero set of Ramanujan's partial theta function $\theta (q,x):=\sum _{j=0}^{\infty}q^{j(j+1)/2}x^j$, $q\in (-1,0)\cup (0,1)$, $x\in \mathbb{R}$. We show that for each $q\in (0,1)$, there exists a line Re$x=-a$, $a\geq 5$, such that all real zeros of $\theta(q,.)$ 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 $\geq -6$. For $q\in (-1,0)$, there are no negative zeros $\geq -2.4$ and no positive zeros $\leq 2.4$, except the smallest one.