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

TL;DR

This work analyzes the complex zeros of the partial theta function $\theta(q,x)=\sum_{j\ge0} q^{j(j+1)/2} x^j$ for $q\in(0,1)$. By relating $\theta$ to the Jacobi theta function via $\Theta^*(q,x)=\Theta(\sqrt{q},\sqrt{q}x)$ and a decomposition $\theta=\Theta^*-G$, the authors develop a contour-based framework (Katsnelson's contour) and spectral analysis to constrain zero locations. They prove that zeros with nonnegative real part lie in a half-annulus $\{\mathrm{Re}\,x\ge0,\ 1<|x|<5\}$, with an outer-radius barrier $e^{\pi/2}$, and show no such zeros for $q\in(0,0.66874\ldots]$; zeros with negative real part are confined to the left open half-disk of radius $49.8$. Additionally, they establish the existence of purely imaginary zeros approaching modulus $e^{\pi/2}$ along a sequence of $q_s$, and provide detailed localization of zeros across spectral intervals. These results connect the zero distribution of $\theta$ to its spectral structure and deepen understanding of its analytic properties relevant to modular and $q$-series contexts.

Abstract

We prove that for any $q\in (0,1)$, all complex conjugate pairs of zeros of the partial theta function $θ(q,x):=\sum _{j=0}^{\infty}q^{j(j+1)/2}x^j$ 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 $e^{π/2}=4.810477382\ldots$, and that for $q\in (0,0.2^{1/4}=0.6687403050\ldots ]$, $θ(q,.)$ 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.

Figures (1)

• Figure 1: Katsnelson's contour (in dashed line) and the borders of the domain $\mathcal{D}$ and the half-annulus $\mathcal{A}$ (in solid line).

Theorems & Definitions (21)

• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4
• Theorem 6
• Lemma 8
• proof
• Proposition 9
• Proposition 10
• proof