Finite-order positivity and growing negative inertia for averaged Riemann theta products
Loading original first page…
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 3 log(N)/(8 pi)-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 3 log(N)/(4 pi)-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. We consolidate, with explicit attribution, 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 ceil(A^(31/4)) in both parities. A classical lognormal counterexample explains why finite moments and a negative density region alone are insufficient.
Read the full paper in the original PDF.
Edition files
- Paper.pdf371.1 KiB · pdf
SHA-256
a0f21b6a7ea2efb46154c21ddb08ac1c65910abc6ce0dc251399a1592787806c - Publication_Package.zip918.1 KiB · package
SHA-256
d4287f0f8cf8968010494d24c94821bc909cb6e6c0b2183ef5bec90c81c7f320
Available editions
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
Common negative subspaces and rank obstructions concern the complete upper-tail form and the broader monotone-weight cone. Constants are conservative. The earlier edition 6 remains a separate attributed predecessor.
Disclosures
Author preprint. AI-assisted research and internal independent mathematical/code review.
Reproduction
The 44-member package contains the 59-page PDF, eleven editable TeX sources, twenty-two Python programs, seven exact-data JSON files, reader instructions, citation metadata and a complete 43-entry checksum manifest. The new eight-node band inputs were freshly regenerated from the complete Mellin evaluator, and normal and optimized runs agree for the band and inertia constant checks. Twenty inherited programs and six inherited data files retain their verified published byte identities. The isolated package-source rebuild reproduces all manuscript page text. Complete analytic proofs, outward finite computations, inherited evidence and cited classification inputs are distinguished. The effective inertia count is conservative and can remain vacuous at very large finite degrees.