{"schema_version":"dataset-v1","id":"zeta-zero-regions","version":1,"title":"Where zeros of ζ can and cannot be","description":"Exact boundaries in the critical strip 0 < Re(s) < 1: the explicit zero-free region σ > 1 − 1/(4.896 log t) for t ≥ 3, the verified height, the proposed half-plane Re(s) > 7/8 and the critical line Re(s) = ½.","license":"CC0-1.0 (values computed from published formulas)","problems":["riemann-hypothesis"],"statements":[{"problem":"riemann-hypothesis","statement":"rh-classical-target","role":"RH asks for every nontrivial zero on Re(s) = ½."},{"problem":"riemann-hypothesis","statement":"rh-finite-height","role":"Every zero with 0 < t ≤ 3,000,175,332,800 is verified on the line."},{"problem":"riemann-hypothesis","statement":"rh-explicit-zero-free-region-2026","role":"No zeros right of the explicit boundary."},{"problem":"riemann-hypothesis","statement":"rh-proposed-seven-eighths-half-plane","role":"Proposed: no zeros with Re(s) > 7/8."}],"figure":{"family":"region-plot","title":"Where zeros can and cannot be","caption":"Height t on a logarithmic scale. Proved regions are solid, the proposed region is dashed. RH concerns every height, above any verified range.","x_label":"Re(s) = σ","y_label":"t","x_range":[0,1],"y_range":[1,100000000000000000000],"y_scale":"log10","x_ticks":[[0,"0"],[0.5,"½"],[0.875,"7/8"],[1,"1"]],"layers":[{"key":"verified","label":"Verified: all zeros on the line, t ≤ 3·10¹²","kind":"band","status":"proved","statement":"rh-finite-height","x":null,"y_max":3000175332800,"series":null},{"key":"zero_free","label":"Zero-free: σ > 1 − 1/(4.896 log t)","kind":"curve_region","status":"proved","statement":"rh-explicit-zero-free-region-2026","x":null,"y_max":null,"series":"zero_free_boundary"},{"key":"proposed","label":"Proposed: no zeros for σ > 7/8","kind":"half_plane","status":"proposed","statement":"rh-proposed-seven-eighths-half-plane","x":0.875,"y_max":null,"series":null},{"key":"target","label":"RH: every zero on σ = ½","kind":"line","status":"target","statement":"rh-classical-target","x":0.5,"y_max":null,"series":null}],"supersedes":"visual-riemann-hypothesis-overview"},"provenance":{"method":"Boundary σ(t) = 1 − 1/(4.896 ln t) evaluated at 201 heights equally spaced in log10 t from log10 3 to 20; other layers are the stated constants.","software":{"python":"3.12.13"},"precision":"Boundary rounded to 6 decimal places.","generator":"openmath/scripts/build-datasets.py","generated_on":"2026-10-20"},"data":{"series":[{"key":"zero_free_boundary","label":"σ = 1 − 1/(4.896 log 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