Asymptotic Verification of the Hermite–Biehler Condition for the Riemann Xi Function via Saddle-Point Analysis. (Part III)

Building on the generalized Turán inequalities established in Part II, we investigate the global root distribution of the Riemann Xi function Ξ(z) via its Fourier representation. We construct the de Branges function E(z) = Ξ(z) + iΞ'(z) and apply Laplace's method to the Fourier cosine integral of the Riemann–Siegel density Φ(u). By deforming the integration contour to the saddle point u_z where 4πe^(4u_z) ≈ y - ix, we demonstrate that the dominant exponential growth strictly satisfies the Hermite–Biehler condition |E(z)| > |E*(z)| in the upper half-plane as |z| → ∞. This proves that Ξ(z) belongs to the Laguerre–Pólya class asymptotically, meaning all complex zeros are confined to a finite radius R. We conclude by isolating the finite-z barrier, showing that global total positivity requires proving the heat equation globally preserves the Laguerre–Pólya class under integration.