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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
2 · 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 O_ε(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 degree d ≥ 1, every integer shift n ≥ max(65536,ceil(16√(d+1)log(d+3))) yields a hyperbolic xi Jensen polynomial; a sharper sufficient scalar test is supplied.

Reproduction notes

The 6-page manuscript and complete editable LaTeX sources are supplied in this 8-member package. README.md gives the build and reproduction commands. Reproducibility code: verify.py. The accompanying manuscript and reader instructions specify the mathematical scope and required dependencies.

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Cite this edition

Scott Schubr. (2026). An explicit hyperbolicity threshold for xi Jensen polynomials (Version 2). Luminary Research Archive. https://research.luminaryoddities.com/explicit-hyperbolicity-threshold-xi-jensen-polynomials/v/2

BibTeX
@techreport{luminary_explicit_hyperbolicity_threshold_xi_jensen_polynomials_v2,
  title = {An explicit hyperbolicity threshold for xi Jensen polynomials},
  author = {Scott Schubr},
  year = {2026},
  institution = {Luminary Research Archive},
  type = {Research Article},
  note = {Version 2},
  url = {https://research.luminaryoddities.com/explicit-hyperbolicity-threshold-xi-jensen-polynomials/v/2}
}

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