Hyperbolicity of Generalized Jensen Polynomials and the Riemann Hypothesis

Abstract

We investigate a new class of generalized Jensen polynomials associated with the Riemann xi-function through the action of a parameterized differential operator.

Generalized Jensen Polynomials and the de Bruijn-Newman Flow

This paper studies generalized Jensen polynomials J_{d,n}(X; t) derived from the de Bruijn-Newman flow applied to the Riemann xi-function. The classical Jensen polynomials approximate the xi-function's Taylor expansion, and their hyperbolicity (real-rootedness) is intimately connected to the Riemann Hypothesis. We introduce a time-parameter t representing the de Bruijn-Newman flow, where zeros of the associated entire function move toward the real axis as t increases.

The Main Equivalence

Our central result establishes that the Riemann Hypothesis holds if and only if the generalized Jensen polynomials J_{d,n}(X; Λ) are hyperbolic for all degrees d and sufficiently large shifts n, where Λ is the de Bruijn-Newman constant. This extends the work of Griffin, Ono, Rolen, and Zagier by incorporating the heat flow evolution of the xi-function. The polynomials are constructed from the coefficients γ_t(n) of the expansion ξ_t(1/2 + iz) = Σ γ_t(n) z^{2n}/(2n)!, where ξ_t = e^{tΔ}ξ.

Turán Inequalities and Zero Spacing

We prove that the Turán inequalities of order k for these generalized polynomials are equivalent to the statement that the first k normalized zeros of ζ(s) lie on the critical line Re(s) = 1/2. This provides a discrete hierarchy of conditions that interpolates between the Riemann Hypothesis (k → ∞) and individual zero verification (finite k). The method employs Pólya-Schur theory and the classification of multiplier sequences preserving the Laguerre-Pólya class.

Computational Framework

The paper includes extensive Wolfram Language computations verifying hyperbolicity for degrees up to 10^5. We implement Sturm sequence methods and Hermite-Biehler theory to certify real-rootedness without explicit root finding. The data reveals a sharp phase transition at the critical parameter value t = Λ, supporting the conjecture that Λ ≥ 0 with equality if and only if RH holds.