A Jensen–Pólya and Bivariate Stability Approach to the Riemann Hypothesis

Abstract

We present a self-contained proof of the Riemann Hypothesis within the FEATAM (From Energy Alone To All Mathematics) framework. Starting from an energetic heat kernel on the integer lattice, we identify the associated energetic zeta function with the classical Riemann zeta function, establish the Euler product and functional equation, and derive the Guinand–Weil explicit formula. Weil’s positivity criterion is recalled as contextual background, while the central argument proceeds independently through a Jensen–Pólya route. We introduce Maclaurin–Jensen polynomials P_D(X) and prove their hyperbolicity for every D ≥ 1. The proof combines strict positivity of the Riemann Ξ-kernel, hyperbolicity of an auxiliary kernel polynomial K_D through the Laguerre–Pólya multiplier sequence c_j = j!/(2j)!, and preservation of bivariate stability under positive-measure integration using Wagner’s continuous-integration theorem. Explicit moment estimates verify the required integrability conditions. The resulting bivariate stability implies that every P_D has only real, strictly negative zeros. We then establish that the associated entire function G has Hadamard order 1/2 and prove local uniform convergence P_D(·/D) → G. The Pólya–Schur closure theorem consequently yields G ∈ LP⁻. Finally, the identity Ξ(x) = G(−x²) implies that every zero of the Riemann Ξ-function is real. Since Ξ(x) = ξ(1/2 + ix), all non-trivial zeros of the Riemann zeta function therefore lie on the critical line Re(s) = 1/2.

Keywords