← Session 27 — ii′-5c/5d: Davenport disc + FE-reflection bounds
thinking (1181 chars)
The user asks "what is next?" — a status/plan question. STATE.md's NEXT ACTION section has the answer. I just wrote it, so I can answer directly from what I know (STATE.md was just updated by me this session). No need to re-read — I have it fresh in context.
Next is session 026: ii′-5e — the partial-fraction identity. Let me lay it out clearly:
1. FIRST ITEM (~0.25 sess): zeroMult(−2n) = 1 for ALL n — generalize ii′-4c's zeroMult_neg_two.
2. ii′-5e proper: the partial-fraction identity
ζ′/ζ(s) = B + Σ_ρ m_ρ(1/(s−ρ) + 1/ρ) − 1/s − 1/(s−1) + ½log π − ½ψ(s/2)
(i) series convergence (mult-weighted Σm/|ρ|² — Zeta23 zero_sum_inv_sq + ii′-1a dominator)
(ii) F := LHS − RHS entire (residue cancellation: nontrivial zeros via ii′-1b machinery; at 1; at 0; at trivial zeros −2n via the first item)
(iii) |F| = o(R) on γ-gap circles ⟹ F constant (subtract F(0), divide by z, max principle + Liouville)
3. Then ii′-5f (tight analysis → A = 10.5, B₀ = 48), ii′-3b (moments), ii′-3c (crux), ii′-6 (assembly), ii′-7 (closure + audit + paper Appendix B).
Keep it concise. This is an informational answer, not a request to start work — answer the question and offer to start.
**Session 026: ii′-5e — the partial-fraction identity** (D-067 §6). This is the pivot sub-stage: the loose 5c/5d bounds become the *growth input* that turns into the identity, and the identity is where the tight 10.5/48 constants live.
The target:
$$\frac{\zeta'}{\zeta}(s) = B + \sum_\rho m_\rho\Big(\frac{1}{s-\rho} + \frac{1}{\rho}\Big) - \frac{1}{s} - \frac{1}{s-1} + \tfrac12\log\pi - \tfrac12\psi(s/2)$$
Three pieces:
1. **Series convergence** — the mult-weighted Σ m_ρ/|ρ|² (Zeta23 `zero_sum_inv_sq` + the ii′-1a dominator).
2. **F := LHS − RHS is entire** — residue cancellation at: nontrivial zeros (ii′-1b's logDeriv machinery), s = 1, s = 0, and the trivial zeros −2n.
3. **|F| = o(R) on γ-gap circles** — the series is O((log R)²) by the dyadic split; |ζ′/ζ| ≤ C(log R)² on the circle by 5c (right) + 5d's FE-reflection (left) + 5d's left-plane digamma bound (ΓR piece) ⟹ F constant (subtract F(0), divide by z, maximum principle + Liouville).
**First item (~0.25 sess): `zeroMult(−2n) = 1` for ALL n** — generalizing ii′-4c's `zeroMult_neg_two` via the product-form Γ recurrence Γ(z) = Γ(z+n+1)/∏_{j≤n}(z+j). Needed for piece 2's entirety at the trivial zeros. Deferred from 5d, queued first in STATE.md.
After 5e: ii′-5f (tight analysis → A = 10.5, B₀ = 48), then ii′-3b (moments), ii′-3c (crux assembly), ii′-6 (L0 assembly), ii′-7 (closure + audit + paper Appendix B + tag upgrade).
Ready to start on `zeroMult(−2n)` whenever you are.
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[{"type":"thinking","thinking":"The user asks \"what is next?\" — a status/plan question. STATE.md's NEXT ACTION section has the answer. I just wrote it, so I can answer directly from what I know (STATE.md was just updated by me this session). No need to re-read — I have it fresh in context.\n\nNext is session 026: ii′-5e — the partial-fraction identity. Let me lay it out clearly:\n\n1. FIRST ITEM (~0.25 sess): zeroMult(−2n) = 1 for ALL n — generalize ii′-4c's zeroMult_neg_two.\n2. ii′-5e proper: the partial-fraction identity\n ζ′/ζ(s) = B + Σ_ρ m_ρ(1/(s−ρ) + 1/ρ) − 1/s − 1/(s−1) + ½log π − ½ψ(s/2)\n (i) series convergence (mult-weighted Σm/|ρ|² — Zeta23 zero_sum_inv_sq + ii′-1a dominator)\n (ii) F := LHS − RHS entire (residue cancellation: nontrivial zeros via ii′-1b machinery; at 1; at 0; at trivial zeros −2n via the first item)\n (iii) |F| = o(R) on γ-gap circles ⟹ F constant (subtract F(0), divide by z, max principle + Liouville)\n3. Then ii′-5f (tight analysis → A = 10.5, B₀ = 48), ii′-3b (moments), ii′-3c (crux), ii′-6 (assembly), ii′-7 (closure + audit + paper Appendix B).\n\nKeep it concise. This is an informational answer, not a request to start work — answer the question and offer to start.","signature":""},{"type":"text","text":"**Session 026: ii′-5e — the partial-fraction identity** (D-067 §6). This is the pivot sub-stage: the loose 5c/5d bounds become the *growth input* that turns into the identity, and the identity is where the tight 10.5/48 constants live.\n\nThe target:\n$$\\frac{\\zeta'}{\\zeta}(s) = B + \\sum_\\rho m_\\rho\\Big(\\frac{1}{s-\\rho} + \\frac{1}{\\rho}\\Big) - \\frac{1}{s} - \\frac{1}{s-1} + \\tfrac12\\log\\pi - \\tfrac12\\psi(s/2)$$\n\nThree pieces:\n1. **Series convergence** — the mult-weighted Σ m_ρ/|ρ|² (Zeta23 `zero_sum_inv_sq` + the ii′-1a dominator).\n2. **F := LHS − RHS is entire** — residue cancellation at: nontrivial zeros (ii′-1b's logDeriv machinery), s = 1, s = 0, and the trivial zeros −2n.\n3. **|F| = o(R) on γ-gap circles** — the series is O((log R)²) by the dyadic split; |ζ′/ζ| ≤ C(log R)² on the circle by 5c (right) + 5d's FE-reflection (left) + 5d's left-plane digamma bound (ΓR piece) ⟹ F constant (subtract F(0), divide by z, maximum principle + Liouville).\n\n**First item (~0.25 sess): `zeroMult(−2n) = 1` for ALL n** — generalizing ii′-4c's `zeroMult_neg_two` via the product-form Γ recurrence Γ(z) = Γ(z+n+1)/∏_{j≤n}(z+j). Needed for piece 2's entirety at the trivial zeros. Deferred from 5d, queued first in STATE.md.\n\nAfter 5e: ii′-5f (tight analysis → A = 10.5, B₀ = 48), then ii′-3b (moments), ii′-3c (crux assembly), ii′-6 (L0 assembly), ii′-7 (closure + audit + paper Appendix B + tag upgrade).\n\nReady to start on `zeroMult(−2n)` whenever you are."}]