A domain free of the zeros of the partial theta function

TL;DR

This paper proves that for all q in (0,1), the partial theta function has no zeros in a specific complex domain and no real zeros greater than or equal to -5, advancing understanding of its zero distribution.

Contribution

It establishes the zero-free regions of the partial theta function in the complex plane and on the real axis for q in (0,1), providing new insights into its zero structure.

Findings

No zeros in the specified complex domain for q in (0,1)

No real zeros greater than or equal to -5

Zero-free regions improve understanding of the partial theta function's behavior

Abstract

We prove that for $q\in (0,1)$, the partial theta function $\theta (q,x):=\sum _{j=0}^{\infty}q^{j(j+1)/2}x^j$ has no zeros in the closed domain $\{ \{ |x|\leq 3\} \cap \{${\rm Re}$x\leq 0\} \cap \{ |${\rm Im}$x|\leq 3/\sqrt{2}\} \} \subset \mathbb{C}$ and no real zeros $\geq -5$.