What question did this study set out to answer?

The aim is to prove the Riemann Hypothesis by establishing properties of its zeros.

May 16, 2026Open Access

Proof of the Riemann Hypothesis via Logarithmic Concavity and the Tope Operator

Key Points

The aim is to prove the Riemann Hypothesis by establishing properties of its zeros.
Established symmetry of zeros using Riemann's functional equation.
Demonstrated uniqueness of the zero on [0,1] via the logarithmic concavity of |ζ|.
Applied the Tope operator for numerical confirmation to 50 decimal places.
Proved symmetry and uniqueness of the zero indicates it must lie at a* = 1/2.
Numerical confirmation provided by the Tope operator with a safety factor ≥ 2×10^7.

Abstract

EN — Version 5.0 — Final proof. The Riemann Hypothesis is established in three steps: (i) symmetry of zeros via Riemann's functional equation; (ii) uniqueness of the zero on 0,1 via logarithmic concavity of |ζ|, combining Hadamard's product formula, von Mangoldt's counting formula, and de la Vallée-Poussin's zero-free region; (iii) pure logic — a unique zero equal to its own reflection must lie at a* = 1/2. The Tope operator provides independent numerical confirmation to 50 decimal places, with a safety factor ≥ 2×10⁷. FR — Version 5.0 — Preuve finale. L'hypothèse de Riemann est établie en trois étapes : symétrie, unicité, logique pure. Le Théorème de Tope fournit une vérification numérique indépendante à 50 décimales avec un facteur de sécurité ≥ 2×10⁷.

Proof of the Riemann Hypothesis via Logarithmic Concavity and the Tope Operator

Key Points

Abstract

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