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OpenAI · 003-01 · 2026-09-30 · Number theory

The Quasi-Riemann Hypothesis: A Zero-Free Half-Plane Re s > 7/8

preprints/The-Quasi-Riemann-Hypothesis-September-30-2026

Family 003. The quasi-Riemann hypothesis.

The authors are OpenAI. This companion does not verify the proof and does not adopt it as a theorem of Ngo Cheung’s series. It is not a premise. O.1 stays NOT_CLOSED. Anti-de Sitter results are not a de Sitter state, not Em02-E’s relational lapse, not XVI-P’s QdS scheme, and not XVI-Q’s sigma-image record. No quantum coefficient is admitted. No compilation is asserted.

Read against the standing problems

Vacuum energy. A zero-free statement or a vacuum phrase is not an explanation of vacuum-energy smallness and not a swampland test. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. Em02-E · XI-A · XVI-P

Proves that every Dirichlet -function, including , is zero-free in , resolving the quasi-Riemann hypothesis. The same half-plane is zero-free for every finite-order Hecke -function over . A companion gives a different proof of the zero-free half-plane .

We prove that all finite-order Hecke L-functions over and all Dirichlet L-functions are zero-free in the half-plane , with the principal pole at s = 1 allowed. In particular, the Riemann zeta function is zero-free in this half-plane, proving the quasi-Riemann hypothesis.

Preprint source and Lean scope

The abstract above is the catalogue abstract. The sections below are the full preprint source, 0 sections. This page is not only the abstract. The manuscript file is loading.

This preprint is named in the Lean scope note for 003: The quasi-Riemann hypothesis. The authors are OpenAI. This companion does not verify the formalization. It is not a premise. The OAI library is not imported. No compilation is asserted. O.1 stays NOT_CLOSED.

QuasiRiemannHypothesis.lean · QuasiRiemannHypothesis.lean

Statement file

DirichletSevenEighths.lean · DirichletSevenEighths.lean
Statement file
HeckeSevenEighths.lean · HeckeSevenEighths.lean
Statement file
SiegelZeros.lean · SiegelZeros.lean
Statement file
  • Vacuum energy
    A zero-free half-plane is not vacuum energy
    Family 003 records a quasi-Riemann zero-free half-plane. That half-plane is not an explanation of vacuum-energy smallness and not a swampland test. The authors are OpenAI. The companion does not verify those proofs.
    Vacuum energyEm02-E exponential potentialXVI-P two-point loop