What question did this study set out to answer?

This work aims to prove the Riemann Hypothesis by establishing conditions that exclude non-trivial zeros off the critical line.

May 21, 2026Open Access

A Proof of the Riemann Hypothesis: Exclusion of Off-Critical-Line Zeros via Series Structure

Puntos clave

This work aims to prove the Riemann Hypothesis by establishing conditions that exclude non-trivial zeros off the critical line.
Utilized the Dirichlet series definition of the Riemann zeta function.
Introduced odd-even splitting to assess the symmetry of term moduli.
Analyzed the uniform decay of term moduli for zero existence on the critical line.
All non-trivial zeros are confined to the critical line Re(s) = 1/2, proving the Riemann Hypothesis.
Demonstrated that off the critical line, the structure of term moduli prevents simultaneous vanishing.

Resumen

Starting from the Dirichlet series definition of the Riemann zeta function, anodd-even splitting is introduced to establish an equivalence condition for the existence of non-trivial zeros. It is then demonstrated that off the critical line (σ = 1/2),the asymmetric structure of term moduli between odd and even parts prevents theodd-even components from simultaneously vanishing. On the critical line (σ = 1/2),however, the uniform decay of term moduli allows complete phase cancellation atspecific t-values, giving rise to zeros. The functional equation guarantees pairedsymmetry of zeros. Consequently, all non-trivial zeros are confined to the criticalline Re(s) = 1/2, thereby proving the Riemann Hypothesis. On this basis, thedetermination of zero positions provides a rigorous foundation for the completeoscillatory behavior of the prime-counting function

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