Short··0:12·Number theory
Quasi-Riemann Hypothesis: The 7/8 Bound
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About this video
The Quasi-Riemann Hypothesis: A Zero-Free Half-Plane Re(s) greater than 7/8
A fixed region with no zeta zeros. OpenAI's manuscript reports a zero-free half-plane Re(s) greater than 7/8 for the Riemann zeta function, all Dirichlet L-functions and finite-order Hecke L-functions over Q(sqrt(-3)). Principal-character poles at s=1 are excluded. The boundary Re(s)=7/8 is not included. This is the quasi-Riemann hypothesis, NOT the full Riemann hypothesis, which remains open according to the manuscript.
Paper: The Quasi-Riemann Hypothesis: A Zero-Free Half-Plane Re(s) greater than 7/8
github.com/openai/math/blob/main/preprints/The-Quasi-Riemann-…
About the animation: A numerical surface of 1.15*log(1+abs(zeta(sigma+i*t))) is sampled on a nonuniform 83 by 378 grid over 0≤sigma≤1.25 and 10≤t≤36 at 25-digit working precision. The grid includes sigma=1/2, sigma=7/8 and the first five positive zero heights. Blue and gold identify the two sides of sigma=7/8; the bright gold boundary is at 7/8, while the reported zero-free region is strictly to its right. A magenta tracer follows the sampled critical-line surface at sigma=1/2 and reaches the five zero markers on measured soundtrack accents. Camera motion, light sweeps and brief screen-visible rings are illustrative. The geometry remains fixed to computed values. This finite plot is not a proof of a global zero-free half-plane or of the full Riemann hypothesis.
GitHub repository: github.com/openai/math
Related formalization scope: github.com/openai/math/blob/main/lean/docs/003.md
The linked scope documents the 7/8 bound and related statements. The formalization was not executed or audited in this production.
Source-reported result. Selected theorem statements and scope were read; this is not an independent proof audit.
'Catalyst' by Scott Buckley - released under CC-BY 4.0. www.scottbuckley.com.au
scottbuckley.com.au/library/catalyst/
Licence: creativecommons.org/licenses/by/4.0/
Music edits: excerpted, crossfaded for looping, and gain adjusted.
About the problem
Number theory·OpenAI Math·Result 003
The quasi-Riemann hypothesis
Proves that every Dirichlet L-function, including ζ(s), is zero-free in Re s > 7/8, resolving the quasi-Riemann hypothesis. The same half-plane is zero-free for every finite-order Hecke L-function over ℚ(√(-3)). A companion gives a different proof of the zero-free half-plane Re s > 11/12. (Lean)
Topics
quasi-Riemann hypothesisRiemann zeta functionzero-free half-planeseven eighths boundDirichlet L-functionsHecke L-functionsanalytic number theoryOpenAI math3D math animation
