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

TL;DR

This paper characterizes the location of complex conjugate zeros of the partial theta function for all q in (0,1), showing they lie within specific half-annuli and disks, with certain regions free of zeros.

Contribution

It provides precise bounds on the regions where zeros of the partial theta function can be found, including the absence of zeros in certain parts of the complex plane.

Findings

Zeros with non-negative real part lie in a half-annulus between radii 1 and 5.

For q in (0, 0.6687], the function has no zeros with non-negative real part.

Zeros with negative real part are confined to a disk of radius 49.8.

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.