Complex spectrum of the partial theta function
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arXiv
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| Format: | Preprint |
| Publié: |
2026
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| _version_ | 1866910271397691392 |
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| author | Shapiro, Boris |
| author_facet | Shapiro, Boris |
| contents | We study the complex spectrum of the partial theta function \[
Θ(q,x)=\sum_{j=0}^{\infty}q^{j(j+1)/2}x^j,
\qquad |q|<1, \] where a spectral value is a parameter for which \(Θ(q,\cdot)\) 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_29991 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Complex spectrum of the partial theta function Shapiro, Boris Classical Analysis and ODEs Complex Variables Primary 33D15, Secondary 30D15, 30E15, 11F27 We study the complex spectrum of the partial theta function \[ Θ(q,x)=\sum_{j=0}^{\infty}q^{j(j+1)/2}x^j, \qquad |q|<1, \] where a spectral value is a parameter for which \(Θ(q,\cdot)\) 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. |
| title | Complex spectrum of the partial theta function |
| topic | Classical Analysis and ODEs Complex Variables Primary 33D15, Secondary 30D15, 30E15, 11F27 |
| url | https://arxiv.org/abs/2605.29991 |