Uniform bounds on locations of zeros of partial theta function

TL;DR

This paper establishes uniform bounds on the locations of zeros of the partial theta function for a range of complex parameters, providing insights into the distribution of zeros as the parameter varies.

Contribution

It proves that for a broad class of parameters, the partial theta function has a predictable number of zeros within specific bounds, extending understanding of its zero distribution.

Findings

Zeros are confined within bounds depending on |q| and n.

Number of zeros with modulus less than |q|^{-n-1/2} is exactly n for large n.

Results hold uniformly for all q in a specified annulus.

Abstract

We consider the partial theta function $\theta (q,z):=\sum _{j=0}^{\infty}q^{j(j+1)/2}z^j$, where $(q,z)\in \mathbb{C}^2$, $|q|<1$. We show that for any $0<\delta _0<\delta <1$, there exists $n_0\in \mathbb{N}$ such that for any $q$ with $\delta _0\leq |q|\leq \delta$ and for any $n\geq n_0$ the function $\theta$ has exactly $n$ zeros with modulus $<|q|^{-n-1/2}$ counted with multiplicity.