Number Theory

    

Proof of the Riemann Hypothesis: A Structural Proof of the Riemann Hypothesis:Well-Ordering, Symmetry, and the Critical Gap Theorem

Authors: Rusin Danilo Olegovich

We present a complete structural proof of the Riemann Hypothesis, based on the interplay between the canonical well-ordering of nontrivial zeros and the symmetry imposed by the functional equation. Working from three established analytic properties of the Riemann zeta function — discreteness of zeros, confinement to the critical strip, and functional equation symmetry — we construct a proof that reduces the Riemann Hypothesis to a purely combinatorial statement about order and symmetry. We prove the Critical Gap Theorem: if a zero off the critical line exists, the first such zero (under the canonical ordering) must lie to the left of the critical line, forcing its functional equation partner to appear later in the ordering. This leads to a contradiction unless no such zero exists. The result is a logically complete, structurally elegant proof of the Riemann Hypothesis requiring no advanced analytic estimates — only classical properties known since the 19th century.

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[v1] 2025-09-06 22:46:10

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