# On 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 q in (-1,1), providing bounds on their real and imaginary parts for different q ranges.

## Contribution

It establishes precise bounds on the complex conjugate zeros of the partial theta function depending on the parameter q, extending understanding of its zero distribution.

## Key findings

- Zeros for q in (0,1) lie within specified bounds in the complex plane.
- Zeros for q in (-1,0) are confined within a particular rectangle.
- The results provide explicit regions where zeros can be located.

## Abstract

We prove that 1) for any $q\in (0,1)$, all complex conjugate pairs of zeros of the partial theta function $\theta (q,x):=\sum _{j=0}^{\infty}q^{j(j+1)/2}x^j$ belong to the set $\{$~Re\,$x\in (-5792.7,0),$~$|$Im\,$x|<132~\}$ $\cup$ $\{ ~|x|<18~\}$ and 2) for any $q\in (-1,0)$, they belong to the rectangle $\{$~$|$Re\,$x|< 364.2,$~$|$Im\,$x|<132~\}$.