Partial Theta Function and Separation in Modulus Property of its Zeros

Abstract

We consider the partial theta function $\theta (q,z):={\sum }_{j=0}^{\infty }q^{j(j+1)/2}z^{j}$, where $z\in \mathbb {C}$ is a variable and $q\in \mathbb {C}$, 0 < |q| < 1, is a parameter. Set $D(a):=\{q\in \mathbb {C}, 0<|q|\leq a,$$\arg (q)\in [\pi /2,3\pi /2]\}$. We show that for $k\in \mathbb {N}$ and q ∈ D(0.55), there exists exactly one zero of θ(q,⋅) (which is a simple one) in the open annulus |q|−k+ 1/2 < z < |q|−k− 1/2 (if k ≥ 2) or in the punctured disk 0 < z < |q|− 3/2 (if k = 1). For k ≠ 2, 3, this holds true for q ∈ D(0.6) as well.