Some analytic properties of the partial theta function

TL;DR

This paper investigates the zero distribution of Ramanujan's partial theta function, establishing new properties of real and complex zeros for various ranges of the parameter q.

Contribution

It proves that for each q in (0,1), all real zeros lie to the left of a certain line, and similar properties hold for q in (-1,0), revealing detailed zero localization.

Findings

All real zeros of $ heta(q,x)$ are to the left of Re$x=-a$ for some a≥5 when q∈(0,1)

For q∈(0,1), no real zeros are greater than or equal to -6

For q∈(-1,0), no negative zeros are ≥ -2.4 and no positive zeros ≤ 2.4, except the smallest one.

Abstract

We prove new properties of the zero set of Ramanujan's partial theta function $\theta (q,x):=\sum _{j=0}^{\infty}q^{j(j+1)/2}x^j$, $q\in (-1,0)\cup (0,1)$, $x\in \mathbb{R}$. We show that for each $q\in (0,1)$, there exists a line Re$x=-a$, $a\geq 5$, such that all real zeros of $\theta(q,.)$ lie to its left and all complex zeros to its right. A similar property is proved for $q\in (-1,0)$. For $q\in (0,1)$, there are no real zeros $\geq -6$. For $q\in (-1,0)$, there are no negative zeros $\geq -2.4$ and no positive zeros $\leq 2.4$, except the smallest one.