What question did this study set out to answer?

This research aims to empirically validate the theoretical framework surrounding Jensen polynomials and their hyperbolicity in the context of the Riemann Hypothesis.

January 14, 2026Open Access

Numerical Verification of Jensen Polynomials Hyperbolicity in Relation to the Riemann Hypothesis

Key Points

This research aims to empirically validate the theoretical framework surrounding Jensen polynomials and their hyperbolicity in the context of the Riemann Hypothesis.
High-precision numerical analysis with 150-digit decimal precision
Investigation of hyperbolic stability for Jensen polynomials up to degree d=20
Utilization of Python-based methodology with the mpmath library
Confirmed hyperbolicity through numerical verification
Achieved consistent alignment of root distributions towards the real axis
Integrated empirical results that support the logical reasoning behind the Riemann Hypothesis

Abstract

This updated version provides a rigorous empirical validation of the theoretical framework proposed in the first version. By employing high-precision numerical analysis (150-digit decimal precision), we investigate the hyperbolic stability of Jensen polynomials associated with the Riemann Xi-function. Key Updates in Version 2.0: Numerical Evidence: Introduction of new computational data for Jensen polynomials up to degree d=20. Python-Based Methodology: Utilization of the mpmath library and normalized coefficient scaling to ensure convergence and minimize ill-conditioning. Root Distribution Analysis: Visual and tabular representation of the maximum imaginary parts of roots, demonstrating consistent alignment towards the real axis. Extended Discussion: Integration of empirical results with the logical reasoning supporting the Riemann Hypothesis.

Numerical Verification of Jensen Polynomials Hyperbolicity in Relation to the Riemann Hypothesis

Key Points

Abstract

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