An Explicit Quintic Obstruction for the Riemann Theta Kernel
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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.
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Research details
- Subjects
- Mathematics · Number Theory and Special Functions
- Keywords
- Riemann theta kernel, Positive definite functions, Hankel matrices, Rational interval arithmetic
- Manuscript date
- 2026-10-13
- Reader reuse
- No additional permission granted
Scope
This result concerns the normalized theta product at the specified fixed center.
Disclosures
Generative AI tools, including OpenAI's Codex, assisted with mathematical exploration and review, preparation of verification code, literature searches, and manuscript drafting and formatting. The author takes responsibility for the manuscript and supplementary materials, including all mathematical claims, computations, and references.
Reproduction
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References and related resources
- 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.