← Session 24 — ii′-2…4: swaps, (4.10) design, contour + website roadmap fixes
thinking (612 chars)
`ZetaUpperBnd`: ∃ A ∈ (0,1/2], C > 0, ∀ σ t, 3 < |t|, σ ∈ [1 − A/log|t|, 2]: ‖ζ(σ+it)‖ ≤ C·log|t|. So Zeta23 has ‖ζ‖ ≤ C log|t| for σ ≥ 1 − A/log|t| — i.e. σ down to ~1 − small. NOT the left-half-plane region. For σ < 1−A/log t (in particular σ ≤ 1/2), we need the FE reflection.
Does Zeta23 have a ζ bound for σ < 1/2 via the FE? PNT repos usually don't need it. Let me grep for the FE usage / riemannZeta_one_sub in Zeta23, and for any χ-bound or Gamma ratio bounds.
Also grep for what StrongPNTPrefix offers (BorelCaratheodoryDeriv, LogOfAnalyticFunction, ZerosBound) and check Mathlib's borelCaratheodory.
`ZetaUpperBnd` covers σ ≥ 1 − A/log|t| only — the σ < 1/2 side needs the FE reflection with a Γ-ratio (Stirling) bound. Checking what exists for that, plus the StrongPNTPrefix machinery.
tool: Bash
input
{
"command": "cd D:/Python/Quantum/zeta-23-lean && grep -rn \"riemannZeta_one_sub\\|Stirling\\|stirling\" Zeta23/ --include=*.lean -r | head -10; grep -rn \"theorem BorelCaratheodoryDeriv\\|theorem LogOfAnalyticFunction\\|theorem ZerosBound\\|def ZerosBound\\|lemma BorelCaratheodoryDeriv\\|lemma LogOfAnalyticFunction\\|lemma ZerosBound\" Zeta23/FromPNTPlus/StrongPNTPrefix.lean | head -8",
"description": "Search Zeta23 for FE/Stirling and StrongPNTPrefix lemma names"
}
raw content_json (database row)
[{"type":"thinking","thinking":"`ZetaUpperBnd`: ∃ A ∈ (0,1/2], C > 0, ∀ σ t, 3 < |t|, σ ∈ [1 − A/log|t|, 2]: ‖ζ(σ+it)‖ ≤ C·log|t|. So Zeta23 has ‖ζ‖ ≤ C log|t| for σ ≥ 1 − A/log|t| — i.e. σ down to ~1 − small. NOT the left-half-plane region. For σ < 1−A/log t (in particular σ ≤ 1/2), we need the FE reflection. \n\nDoes Zeta23 have a ζ bound for σ < 1/2 via the FE? PNT repos usually don't need it. Let me grep for the FE usage / riemannZeta_one_sub in Zeta23, and for any χ-bound or Gamma ratio bounds.\n\nAlso grep for what StrongPNTPrefix offers (BorelCaratheodoryDeriv, LogOfAnalyticFunction, ZerosBound) and check Mathlib's borelCaratheodory.","signature":""},{"type":"text","text":"`ZetaUpperBnd` covers σ ≥ 1 − A/log|t| only — the σ < 1/2 side needs the FE reflection with a Γ-ratio (Stirling) bound. Checking what exists for that, plus the StrongPNTPrefix machinery."},{"type":"tool_use","id":"Bash_157","name":"Bash","input":{"command":"cd D:/Python/Quantum/zeta-23-lean && grep -rn \"riemannZeta_one_sub\\|Stirling\\|stirling\" Zeta23/ --include=*.lean -r | head -10; grep -rn \"theorem BorelCaratheodoryDeriv\\|theorem LogOfAnalyticFunction\\|theorem ZerosBound\\|def ZerosBound\\|lemma BorelCaratheodoryDeriv\\|lemma LogOfAnalyticFunction\\|lemma ZerosBound\" Zeta23/FromPNTPlus/StrongPNTPrefix.lean | head -8","description":"Search Zeta23 for FE/Stirling and StrongPNTPrefix lemma names"},"extra_content":null}]