"A Proof of the Riemann Hypothesis via the Barner Potential and Lambert W Function

This preprint presents a proof that all non-trivial zeros of the Riemann zeta function ζ (s) satisfy Re (s) = 1/2 (the Riemann Hypothesis). The approach is based on the Barner potential J (ρ) = Σ_γ log|ρ− (1/2+iγ) | + Jₐrch (ρ), a real-valued function whose singularities coincide exactly with the zeros of ζ. The proof establishes three independent results using classical analytic tools (Binet series for the digamma function, Lambert W function, Barner–Guinand regularization of the Weil explicit formula): 1. The second partial derivative ∂²J/∂σ² is strictly positive throughout the critical strip (0, 1) × (14. 135, ∞), proved analytically in two cases via the Binet series and the monotonicity of ImW (σ+it). 2. A logarithmic pole of J at σ₀ ≠ 1/2 is incompatible with ∂²J/∂σ² > 0, by a direct computation showing ∂²J/∂σ² → −∞ in the horizontal approach to any off-line pole. 3. No non-trivial zero of ζ exists for |Im (s) | ≤ 14. 135, as established computationally. Together these steps imply that no zero of ζ can lie off the critical line, for any imaginary part. The paper includes a corollary showing that the functional equation ξ (s) = ξ (1−s) independently excludes entire zero quartets off the critical line.