Research Article · Preprint Mathematics
Three-point positivity for weighted products of Riemann theta density powers
- Primary subject
- Mathematics
- Subject subfields
- Classical and Real Analysis
- Paper type
- Research Article
- Manuscript stage
- Preprint
- Manuscript
- Archived
- Edition
- 2 · Permanent link
Overview
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.
Keywords theta density · three-point positivity · positive definite kernels · curvature · weighted products
Scope of the work
Strict three-point translation positivity for nonnegative center mixtures of products of Riemann theta density powers for every exponent at least 5/8.
Reproduction notes
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.
The manuscript
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Author 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.
A stable scholarly record
Cite this edition
Scott Schubr. (2026). Three-point positivity for weighted products of Riemann theta density powers (Version 2). Luminary Research Archive. https://research.luminaryoddities.com/three-point-positivity-for-weighted-riemann-theta-products/v/2
BibTeX
@techreport{luminary_three_point_positivity_for_weighted_riemann_theta_products_v2,
title = {Three-point positivity for weighted products of Riemann theta density powers},
author = {Scott Schubr},
year = {2026},
institution = {Luminary Research Archive},
type = {Research Article},
note = {Version 2},
url = {https://research.luminaryoddities.com/three-point-positivity-for-weighted-riemann-theta-products/v/2}
}Loading discussion…