An explicit hyperbolicity threshold for xi Jensen polynomials
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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.
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Research details
- Subjects
- Mathematics · Number Theory · Complex Analysis
- Keywords
- Riemann xi-function, Jensen polynomials, hyperbolicity, entire functions, effective bounds
- Manuscript date
- 2026-10-25
- Reader reuse
- No additional permission granted
Scope
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.
Disclosures
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Reproduction
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.