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 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.
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558ed68b771ffe7c6cf54a0b07888196e11df663e5facde6075ee46c9bc751e3 - Xi_Jensen_threshold_publication.zip89.3 KiB · package
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3208a508d87a40fe9eba343e75916525ff4bc0ca2a75f9ae5f8c363cee74399d
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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 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.
Disclosures
AI-assisted mathematical exploration, manuscript preparation and computational checks. The author takes responsibility for the content.
Reproduction
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.