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Research Article · Preprint Mathematics

An Explicit Quintic Obstruction for the Riemann Theta Kernel

Scott Schubr

Luminary Oddities LLC

  • Mathematics
  • Number Theory and Special Functions
Primary subject
Mathematics
Subject subfield
Number Theory and Special Functions
Paper type
Research Article
Manuscript stage
Preprint
Manuscript
Archived
Edition
1 · Permanent link
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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 s₀ = ¼ log(13/(4π)), the polynomial P(X) = X⁵ − 264X³ + 10952X satisfies ℒₛ₀(P²) ≤ −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

This result concerns the normalized theta product at the specified fixed center. It does not determine positive definiteness of kernels obtained by averaging over the center.

Reproduction notes

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References and related publications

  1. B. Rodgers and T. Tao. The de Bruijn–Newman constant is non-negative. Forum of Mathematics, Pi 8 (2020), e6.
  2. G. Csordas. Fourier transforms of positive definite kernels and the Riemann xi-Function. arXiv:1309.0055v2 (2014).
  3. R. L. Hudson. When is the Wigner quasi-probability density non-negative? Reports on Mathematical Physics 6 (1974), 249–252.
  4. 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 the Riemann Theta Kernel (Version 1). Luminary Research Archive. https://research.luminaryoddities.com/explicit-quintic-obstruction-riemann-theta-kernel/v/1

BibTeX
@techreport{luminary_explicit_quintic_obstruction_riemann_theta_kernel_v1,
  title = {An Explicit Quintic Obstruction for the Riemann Theta Kernel},
  author = {Scott Schubr},
  year = {2026},
  institution = {Luminary Research Archive},
  type = {Research Article},
  note = {Version 1},
  url = {https://research.luminaryoddities.com/explicit-quintic-obstruction-riemann-theta-kernel/v/1}
}

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