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This research explores the behavior of off-line zeros of the Riemann zeta function and their implications for the Riemann Hypothesis.

April 16, 2026Open Access

Off-Line Zero Instability and a Conditional Reduction of the Riemann Hypothesis via Density Estimates

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This research explores the behavior of off-line zeros of the Riemann zeta function and their implications for the Riemann Hypothesis.
Developed the Off-Line Instability Theorem for off-line zeros of the Riemann zeta function.
Analyzed the tension function Tau(sigma, T) to establish divergence conditions.
Examined the relationship between off-line zeros and their density estimates.
Proved that for sigma > 29/34, the tension function Tau(sigma, T) diverges as T increases.
Demonstrated that off-line zeros far from the critical line have self-defeating structural properties.
Established a conditional reduction of the Riemann Hypothesis relying on the Molina Spectral Instability Conjecture.

Abstract

We establish an unconditional instability theorem for hypothetical off-line zeros of the Riemann zeta function. The Off-Line Instability Theorem proves that for sigma > 29/34, the tension function Tau (sigma, T) = T^sigma - 1/2 / N (sigma, T) diverges as T increases, where N (sigma, T) counts zeros with Re (rho) > sigma and Im (rho) 1/2), then the Riemann Hypothesis is true via the integrality of the counting function. This reduction is strictly weaker than RH and reduces to improved zero density estimates. Part of the Formulametrics Research Program.

Off-Line Zero Instability and a Conditional Reduction of the Riemann Hypothesis via Density Estimates

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