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
- 1 · Permanent link
Overview
Abstract
Let F(z)=xi((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, without establishing its density condition. A singular-inner countermodel explains why fixed-width decay and interior zero jets alone do not imply a general synthesis theorem. The fixed-boundary problem at arbitrary 0<y<=1, a uniform target-class degree bound, original-form sign and the Riemann hypothesis remain unresolved.
Keywords polynomial approximation · Riemann xi-function · spectral synthesis · zero ideals · Meixner-Pollaczek polynomials · holomorphic Dirichlet spaces
Scope of the work
Same-boundary synthesis requires y>1; synthesis with a strict strip margin covers all positive widths. The fixed-Hilbert and projective closures equal the full zero ideal. The resulting F-dependent RH-equivalent density condition is not established. No original-form sign, universal degree rate, numerical onset or RH proof is claimed.
Reproduction notes
The complete publication package includes all editable LaTeX inputs, a portable build script and an exact coefficient verifier. Finite controls corroborate algebra; the analytic synthesis and zero-ideal conclusions are established by the manuscript 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 1). Luminary Research Archive. https://research.luminaryoddities.com/polynomial-synthesis-zero-ideals-riemann-xi-function/v/1
BibTeX
@techreport{luminary_polynomial_synthesis_zero_ideals_riemann_xi_function_v1,
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 1},
url = {https://research.luminaryoddities.com/polynomial-synthesis-zero-ideals-riemann-xi-function/v/1}
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