Three-point positivity for weighted Riemann theta products
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Abstract
We give an explicit curvature argument for three-point positivity of weighted products of the Riemann theta density. For every alpha at least 5/8 and every nonnegative Borel center measure with a finite positive integral of Phi^(2alpha), the averaged product Phi(s+t/2)^alpha Phi(s-t/2)^alpha has positive definite three-point translation matrices at distinct real positions. Repeated positions give positive semidefinite matrices. A general curvature criterion combines with positive-Laplace estimates for the complete theta-series tail; all infinite-tail bounds and rational interval estimates are explicit.
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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
For the complete positive even Riemann theta density Phi, every alpha>=5/8 and every nonnegative Borel center measure nu with 0<integral Phi^(2alpha)dnu<infinity give strict positive-definite translation matrices on every distinct real triple for H(t)=integral Phi(s+t/2)^alpha Phi(s-t/2)^alpha dnu(s); repetitions PSD. General sufficient criterion k=-(log phi)second>0 and0<=ksecond<=2k^2 proved on R. Correct derivative-normalized y=0 limit included. Three-point only; density powers before averaging; alpha5/8 sufficient not optimal. Umihara qualitative sign and classical angle geometry credited. Odd-quintic and rank-five results not package members.
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
The approved seven-member package includes the seven-page manuscript, editable LaTeX source, exact symbolic controls, requirements, citation metadata, README and complete public checksums. Follow the README for reader rebuilding and normal/optimized verification. Finite exact controls support the analytic infinite-tail argument; internal review is distinct from external human refereeing.