High Precision Verification of the Turán Inequalities for the Riemann Xi Function up to n=1000 (Part I)

The Riemann Hypothesis is equivalent to the statement that the Taylor coefficients a_n of the Riemann Xi function Ξ(t) form a Pólya Frequency sequence. We present a high precision numerical verification of the Turán inequalities a_n² ≥ aₙ₋₁aₙ₊₁ for this sequence up to n=1000. Using a dynamically bounded Riemann Siegel quadrature method at 400 decimal digits of precision, we bypass the catastrophic cancellation that plagues standard contour integration for high indices. We compute the Turán quotients q_n = (aₙ₋₁aₙ₊₁) / a_n², demonstrating strict log concavity (q_n ≤ 1) and smooth monotonic convergence toward 1. The numerical evidence is consistent with the asymptotic behavior 1 - q_n ~ C/n, where the estimated constant C_n stabilizes near 1.66 with expected logarithmic corrections. These results provide the strongest empirical support to date for the Laguerre Pólya property of the Riemann Xi function.