Research Article · Preprint Mathematics
Polynomial synthesis and zero ideals for the Riemann xi-function
- Primary subject
- Mathematics
- Subject subfields
- Complex Analysis · Functional Analysis · Number Theory and Special Functions
- Paper type
- Research Article
- Manuscript stage
- Preprint
- Manuscript
- Archived
- Edition
- 2 · Permanent link
Overview
Abstract
Let F(z)=ξ((1+iz)/2)/4. On every fixed horizontal strip of half-width y>1, we approximate arbitrary holomorphic zero-compatible Dirichlet targets by ordinary polynomial multiples of F in the same boundary norm, without assuming a larger-strip extension. For all positive widths we also prove synthesis with a strict strip margin. Convergence retains the derivative energy of the entire product, an anchored value, and the original real-axis absolute-potential weights. A paired-product minimum-modulus estimate controls initially large quotients; Gaussian regularization, Fourier truncation and a Meixner–Pollaczek expansion then give a constructive return to polynomials. The zero-free collars outside the unconditional zero strip permit a same-width Sobolev Fourier passage. In one fixed Hilbert space at each y>1, and in the compatible projective space over all finite strips, the closure of the polynomial multiples is exactly the full zero ideal. The potential weights themselves force cancellation at real zeros. The remaining quotient has complex dimension equal to total nonreal zero multiplicity when that number is finite, and infinite dimension otherwise. This gives an F-dependent density characterization equivalent to the Riemann hypothesis. A singular-inner countermodel explains why fixed-width decay and interior zero jets alone do not imply a general synthesis theorem.
Keywords polynomial approximation · Riemann xi-function · spectral synthesis · zero ideals · Meixner-Pollaczek polynomials · holomorphic Dirichlet spaces
Scope of the work
Same-width zero-compatible polynomial synthesis for the specified Riemann-xi multiplier at strip half-width greater than one, with energy/weight control, full zero-ideal closure and its scoped RH density equivalence.
Reproduction notes
13-page manuscript and 13 source-package members. Use python build.py with Tectonic to compile main.tex and the included sections. Standard-library python verify_coefficients.py checks the finite algebraic coefficient controls accompanying the analytic synthesis proofs.
The manuscript
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Author disclosures
AI tools assisted with mathematical development, literature comparison, proof checking and manuscript preparation. The author is responsible for the statements and proofs. This preprint does not claim external peer review or formal proof verification.
A stable scholarly record
Cite this edition
Scott Schubr. (2026). Polynomial synthesis and zero ideals for the Riemann xi-function (Version 2). Luminary Research Archive. https://research.luminaryoddities.com/polynomial-synthesis-zero-ideals-riemann-xi-function/v/2
BibTeX
@techreport{luminary_polynomial_synthesis_zero_ideals_riemann_xi_function_v2,
title = {Polynomial synthesis and zero ideals for the Riemann xi-function},
author = {Scott Schubr},
year = {2026},
institution = {Luminary Research Archive},
type = {Research Article},
note = {Version 2},
url = {https://research.luminaryoddities.com/polynomial-synthesis-zero-ideals-riemann-xi-function/v/2}
}Loading discussion…