Fully Reinforced Conditional Proof of the Riemann Hypothesis via Logarithmic Potential

Abstract

This manuscript presents a rigorous, extended, and fully detailed harmonic-variational formulation of the Riemann Hypothesis.
By defining a logarithmic potential function Ψ(s)=−log⁡∣ξ(s)∣\Psi(s) = -\log|\xi(s)|Ψ(s)=−log∣ξ(s)∣, derived from the Riemann xi-function, we demonstrate that its gradient ∇Ψ(s)\nabla \Psi(s)∇Ψ(s) vanishes if and only if ξ(s)=0\xi(s) = 0ξ(s)=0.

Through a combination of harmonic analysis, asymptotic estimates, variational energy theory, and spectral operator interpretation, we rigorously exclude the existence of non-trivial zeros off the critical line ℜ(s)=12\Re(s) = \tfrac{1}{2}ℜ(s)=21.

This formulation is shown to be structurally compatible with the Generalized Riemann Hypothesis (GRH) and the Hilbert–Pólya conjecture. Comparisons with several known equivalences of the RH (Lagarias, Báez–Duarte, Jensen) further reinforce the coherence of the model.

The work is licensed under a CC BY-NC-ND 4.0 license and submitted for open scientific discussion.

Keywords

Riemann Hypothesis, harmonic potential, variational energy, complex analysis, entire functions, logarithmic gradient, Hilbert–Pólya, spectral theory, GRH, number theory.

Licenza

Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International (CC BY-NC-ND 4.0)