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

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) < T. Off-line zeros far from the critical line are structurally self-defeating: their prime counting oscillation grows faster than their allowed density. The threshold sigma = 29/34 is arithmetically determined by the Ingham-Huxley density estimate. A conditional reduction follows: if the Molina Spectral Instability Conjecture holds (divergence for all sigma > 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.

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Publication Details

Journal
Zenodo (CERN European Organization for Nuclear Research)
Published
2026-04-14
DOI
https://doi.org/10.5281/zenodo.19563899
Primary Topic
Analytic Number Theory Research
Type
article
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article

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

Juan Gabriel Molina
Zenodo (CERN European Organization for Nuclear Research)
Analytic Number Theory Research
article

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

Juan Gabriel Molina
article en

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) < T. Off-line zeros far from the critical line are structurally self-defeating: their prime counting oscillation grows faster than their allowed density. The threshold sigma = 29/34 is arithmetically determined by the Ingham-Huxley density estimate. A conditional reduction follows: if the Molina Spectral Instability Conjecture holds (divergence for all sigma > 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.

Zenodo (CERN European Organization for Nuclear Research)
Openalex Percentile: Top 4%
Analytic Number Theory Research
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