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Abstract:We study the complex spectrum of the partial theta function \[
\Theta(q,x)=\sum_{j=0}^{\infty}q^{j(j+1)/2}x^j,
\qquad |q|<1, \] where a spectral value is a parameter for which \(\Theta(q,\cdot)\) has a multiple zero. Since the function is defined here only for \(|q|<1\), all spectral values are strictly inside the unit disk; boundary points on \(|q|=1\) occur only as accumulation points of the spectrum. The paper combines two complementary points of view. Near the unit circle we prove that every point of \(|q|=1\) is an accumulation point of the spectrum; the proof uses explicit spectral factors of truncations, the Jacobi triple product, and a boundary-window lifting argument near roots of unity. Inside a fixed subdisk, illustrated for \(|q|\leq 0.8\), the true spectrum is locally finite and must be separated carefully from the much larger branch loci of truncations and Jensen polynomials. We give a truncation-seeded Newton procedure which produces a discrete list of candidate spectral values, explain the caustic/escaping-root mechanism in finite approximants, and record numerical monodromy experiments using a radial convention: for a spectral point \(q_*\), roots are labelled at the point \(0.1q_*/|q_*|\) on the small circle and then continued along the straight radial segment to \(q_*\). This convention gives a coherent set of collision labels in the disk, treats negative real spectral values from the base point \(-0.1\), and leads to a preliminary rational-direction heuristic for radial monodromy.

Submission history

From: Boris Shapiro [view email]
[v1] Thu, 28 May 2026 14:24:37 UTC (206 KB)
[v2] Fri, 29 May 2026 07:10:02 UTC (206 KB)