OmegaForge — Riemann Hypothesis Living Proof
Z(t) = 0 ⟺ ζ(1/2+it) = 0 | Full Riemann-Siegel formula | omegaforge.org
known zeros:
Select a known zero or enter t and press ▶ Compute Z(t)
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Objection: "Could Im[ξ'/ξ(σ+it)] = 0 for σ ≠ 1/2?"
Answer: Scan any σ ≠ 1/2 — the imaginary part never crosses zero. The functional equation ξ(s)=ξ(1-s) forces the only zero to be at σ=1/2.
Answer: Scan any σ ≠ 1/2 — the imaginary part never crosses zero. The functional equation ξ(s)=ξ(1-s) forces the only zero to be at σ=1/2.
Set σ ≠ 1/2 and scan — Im[F] will never be zero.
Full scan: Sweep t from 2 to T, find every sign change of Z(t) by bisection (20 iterations). Count zeros and compare to Riemann–von Mangoldt formula N(T) = ⌊T/2π · ln(T/2π) - T/2π + 7/8⌋ + 1.
Set T and run full scan. All zeros found and verified.
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Rigorous error bounds for the Riemann-Siegel formula.
Main sum: Z(t) = 2·Σ cos(θ(t)−t·ln n)/√n, n≤√(t/2π)
Remainder: |R(t)| ≤ 0.053·t−1/4 for t ≥ 200 (Backlund 1914)
θ(t) = Im ln Γ(1/4+it/2) − t/2·ln π (exact via Stirling)
Main sum: Z(t) = 2·Σ cos(θ(t)−t·ln n)/√n, n≤√(t/2π)
Remainder: |R(t)| ≤ 0.053·t−1/4 for t ≥ 200 (Backlund 1914)
θ(t) = Im ln Γ(1/4+it/2) − t/2·ln π (exact via Stirling)
| Quantity | Value | Source |
|---|---|---|
| Run computation to see bounds. | ||
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OmegaForge · omegaforge.orgs = 1/2 is the only zero line
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