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An explicit hyperbolicity threshold for xi Jensen polynomials

Scott Schubr

  • Mathematics
  • Number Theory
  • Complex Analysis
Primary subject
Mathematics
Subject subfields
Number Theory · Complex Analysis
Paper type
Research Article
Manuscript stage
Preprint
Manuscript
Archived
Edition
1 · Permanent link
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Overview

Abstract

We give an explicit sufficient shift for hyperbolicity of the degree-d Jensen polynomials attached to Riemann's xi-function. Every integer n ≥ max{65536, ⌈16√(d+1) log(d+3)⌉} is sufficient. The estimate replaces the exponent loss in the qualitative Oε(d^(1/2+ε)) threshold by a critical logarithmic factor and an effective onset. The proof quantitatively specializes the derivative-escape argument of Kim and Lee, using the classical zero strip, Gontcharoff interpolation, Cauchy's estimate and an elementary growth bound for xi. We also give a sharper scalar criterion and a portable exact checker for its rational enclosures.

Keywords Riemann xi-function · Jensen polynomials · hyperbolicity · entire functions · effective bounds

Scope of the work

For the genuine xi Taylor-coefficient family, the note proves the displayed sufficient threshold for every degree d ≥ 1 and every integer shift at or above it. Multiplicities are allowed, and the sharper scalar criterion is also proved.

Reproduction notes

The supplementary package includes the complete TeX source, standard-library Python checker, deterministic scalar output, citation metadata and build instructions. Run python verify.py and python -O verify.py; compile with Tectonic using build.py. The analytic proof establishes the uniform degree and shift quantifiers.

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Author disclosures

AI-assisted mathematical exploration, manuscript preparation and computational checks. The author takes responsibility for the content.

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Cite this edition

Scott Schubr. (2026). An explicit hyperbolicity threshold for xi Jensen polynomials (Version 1). Luminary Research Archive. https://research.luminaryoddities.com/explicit-hyperbolicity-threshold-xi-jensen-polynomials/v/1

BibTeX
@techreport{luminary_explicit_hyperbolicity_threshold_xi_jensen_polynomials_v1,
  title = {An explicit hyperbolicity threshold for xi Jensen polynomials},
  author = {Scott Schubr},
  year = {2026},
  institution = {Luminary Research Archive},
  type = {Research Article},
  note = {Version 1},
  url = {https://research.luminaryoddities.com/explicit-hyperbolicity-threshold-xi-jensen-polynomials/v/1}
}

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