Three-point positivity for weighted products of Riemann theta density powers
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Abstract
We give an explicit curvature argument for three-point positivity of weighted products of the Riemann theta density. If Φ is the usual positive even density, then for every α≥5/8 and every nonnegative measure ν with 0<∫Φ(s)^(2α) dν(s)<∞, the function ∫Φ(s+t/2)^αΦ(s-t/2)^α dν(s) has positive definite three-point translation matrices at distinct real positions. Repeated positions give positive semidefinite matrices. The proof combines a general curvature criterion with positive-Laplace estimates for the complete theta-series tail. All infinite-tail bounds and rational interval estimates are provided explicitly. The result concerns three-point matrices.
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Research details
- Subjects
- Mathematics · Classical and Real Analysis
- Keywords
- theta density, three-point positivity, positive definite kernels, curvature, weighted products
- Manuscript date
- 2026-10-24
- Reader reuse
- No additional permission granted
Scope
Strict three-point translation positivity for nonnegative center mixtures of products of Riemann theta density powers for every exponent at least 5/8.
Disclosures
AI-assisted tools supported research exploration, literature review, manuscript preparation and internal verification. The author retains responsibility for the work. Internal AI-assisted and computational checks are distinct from external human peer review.
Reproduction
7-page manuscript and 7 package members. Compile main.tex with Tectonic or compatible LaTeX. Install the pinned SymPy 1.14.0 requirement and run python exact-controls.py and python -O exact-controls.py for exact symbolic/rational curvature and determinant identities. The complete infinite-tail arguments are in the manuscript.