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| Format: | Recurso digital |
| Language: | English |
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Zenodo
2026
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| Online Access: | https://doi.org/10.5281/zenodo.19230304 |
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- <h1><strong>Zenodo Submission — Metadata</strong></h1> <p><em>A Proof of the Riemann Hypothesis via the Barner Potential and Lambert W Function</em></p> <h2>Title</h2> <p>A Proof of the Riemann Hypothesis via the Barner Potential and Lambert W Function</p> <h2>Authors</h2> <p>Stefano Rivis</p> <h2>Resource Type</h2> <p>Publication → Preprint</p> <h2>License</h2> <p>Creative Commons Attribution 4.0 International (CC BY 4.0)</p> <h2>Keywords</h2> <p>Riemann Hypothesis, Barner potential, Lambert W function, Binet series, Weil explicit formula, zeta function, critical line</p> <h2>MSC 2020</h2> <p>11M26 (Nonreal zeros of ζ(s) and L-functions) • 30D35 (Value distribution of meromorphic functions) • 33E20 (Functions defined by series and integrals)</p> <h2>Description</h2> <p>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).</p> <p>The approach is based on the Barner potential J(ρ) = Σ_γ log|ρ−(1/2+iγ)| + J_arch(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):</p> <p><strong>1. </strong>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 Im[W(σ+it)].</p> <p><strong>2. </strong>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.</p> <p><strong>3. </strong>No non-trivial zero of ζ exists for |Im(s)| ≤ 14.135, as established computationally.</p> <p>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.</p> <p> </p>