Research Article · Preprint Mathematics
An explicit quintic obstruction for a fixed-center Riemann theta product
- Primary subject
- Mathematics
- Subject subfields
- Number Theory and Special Functions
- Paper type
- Research Article
- Manuscript stage
- Preprint
- Manuscript
- Archived
- Edition
- 2 · Permanent link
Overview
Abstract
We exhibit an explicit polynomial square on which the signed derivative functional of a fixed-center Riemann theta product is negative. At the center s_0=(1)/(4)log(13/(4π)), the polynomial P(X)=X^5-264X^3+10952X satisfies ℒ_(s_0)(P^2)≤-1 351 625 671. The certificate uses rational interval arithmetic for derivatives through order ten and an explicit bound on the complete infinite theta tail. It follows that the normalized product at this center is not positive definite and that its odd derivative Hankel block is indefinite.
Keywords Riemann theta kernel · Positive definite functions · Hankel matrices · Rational interval arithmetic
Scope of the work
An explicit fixed-center quintic polynomial-square certificate proves failure of positive definiteness of the specified normalized theta product and indefiniteness of its odd derivative Hankel block.
Reproduction notes
The paper has 6 pages. The editable-source archive contains 11 files. Compile Quintic_Theta_Witness_Publication.tex. The two standard-library verifiers independently regenerate the 160-bit and 192-bit rational certificate calculations, with assertions enabled.
The manuscript
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References and related publications
- B. Rodgers and T. Tao. The de Bruijn–Newman constant is non-negative. Forum of Mathematics, Pi 8 (2020), e6.
- G. Csordas. Fourier transforms of positive definite kernels and the Riemann xi-Function. arXiv:1309.0055v2 (2014).
- R. L. Hudson. When is the Wigner quasi-probability density non-negative? Reports on Mathematical Physics 6 (1974), 249–252.
- Misty | PIRarchive. Where the RH Program Stands: Computing the Shorted Boundary Form. June 2, 2026, Section 6. Informal online numerical report; accessed October 13, 2026.
A stable scholarly record
Cite this edition
Scott Schubr. (2026). An explicit quintic obstruction for a fixed-center Riemann theta product (Version 2). Luminary Research Archive. https://research.luminaryoddities.com/explicit-quintic-obstruction-riemann-theta-kernel/v/2
BibTeX
@techreport{luminary_explicit_quintic_obstruction_riemann_theta_kernel_v2,
title = {An explicit quintic obstruction for a fixed-center Riemann theta product},
author = {Scott Schubr},
year = {2026},
institution = {Luminary Research Archive},
type = {Research Article},
note = {Version 2},
url = {https://research.luminaryoddities.com/explicit-quintic-obstruction-riemann-theta-kernel/v/2}
}Loading discussion…