Finite-order positivity and growing negative inertia for averaged Riemann theta products
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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.
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Research details
- Subjects
- Mathematics · Linear Algebra and Matrix Inequalities · Number Theory and Special Functions · Analysis and Spectral Theory
- Keywords
- Riemann theta density, negative inertia, moment matrices, Meixner-Pollaczek polynomials, time-band concentration, finite-rank repair
- Manuscript date
- 2026-10-24
- Reader reuse
- No additional permission granted
Scope
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.
Disclosures
Author preprint. AI-assisted research and internal independent mathematical/code review; not external human refereeing or formal proof-assistant verification.
Reproduction
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.