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Research Article · Preprint Mathematics

Finite-order positivity and growing negative inertia for averaged Riemann theta products

Scott Schubr

  • Mathematics
  • Analysis and Spectral Theory
  • Number Theory and Special Functions
  • Linear Algebra and Matrix Inequalities
Primary subject
Mathematics
Paper type
Research Article
Manuscript stage
Preprint
Manuscript
Archived
Edition
2 · Permanent link
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Overview

Abstract

We study the contrast between finite-order positivity and all-order failure for averaged Riemann theta products. Our main new theorem proves that, for the complete upper-center-tail form at the exact cutoff A=23/2, each polynomial parity through degree N has negative inertia at least 3log N/(8π)-O(log log N). A fully explicit conservative bound constructs common negative subspaces, with every theta term and approximation error retained. Consequently a correction making the full form positive semidefinite must have rank at least 3log N/(4π)-O(log log N), even if the correction changes with N or couples parities. The proof combines classical sinc concentration with gamma-reference cancellation and uniform Meixner–Pollaczek projection. To place the result in its finite-order context, we consolidate previously established positivity through degree eight, degree-dependent terminal positivity, exact rational negative witnesses of degrees 4096 and 4097, and eventual witness degree at most ⌈ A^(31/4)⌉ in both parities. Reused results are identified explicitly. A classical lognormal counterexample explains why finite moments and a negative density region alone are insufficient.

Keywords Riemann theta density · negative inertia · moment matrices · Meixner-Pollaczek polynomials · time-band concentration · finite-rank repair

Scope of the work

Explicit logarithmically growing negative inertia in each parity for the complete theta upper-center-tail form at cutoff 23/2, and necessary correction-rank growth, alongside identified finite-order predecessors.

Reproduction notes

59-page manuscript and 44 source-package members. Compile main.tex with its included sections. Install python-flint==0.9.0 and run verify_theta_negative_band.py and verify_theta_inertia.py for the new inertia certificates; --recompute-nodes regenerates the interval nodes. The README separately documents all inherited finite-order, cutoff and polynomial-witness checkers and their precision/grid options.

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Author disclosures

Author preprint. AI-assisted research and internal independent mathematical/code review; not external human refereeing or formal proof-assistant verification.

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Cite this edition

Scott Schubr. (2026). Finite-order positivity and growing negative inertia for averaged Riemann theta products (Version 2). Luminary Research Archive. https://research.luminaryoddities.com/finite-order-positivity-growing-negative-inertia-averaged-riemann-theta-products/v/2

BibTeX
@techreport{luminary_finite_order_positivity_growing_negative_inertia_averaged_riemann_theta_products_v2,
  title = {Finite-order positivity and growing negative inertia for averaged Riemann theta products},
  author = {Scott Schubr},
  year = {2026},
  institution = {Luminary Research Archive},
  type = {Research Article},
  note = {Version 2},
  url = {https://research.luminaryoddities.com/finite-order-positivity-growing-negative-inertia-averaged-riemann-theta-products/v/2}
}

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