[{"type":"thinking","thinking":"No Weierstrass product for Gamma in Mathlib's Gamma folder (Lindemann hits are irrelevant). No Hurwitz-at-1/2 direct.\n\nHmm. So the archimedean coefficient extraction for j ≥ 1 needs either:\n1. The digamma partial fraction series (4.19) — built from... the Gamma function's definition in Mathlib. Mathlib's Complex.Gamma — what IS the definition? Let me check Basic.lean — I recall Mathlib defines Complex.Gamma via an entire-function quotient: `Gamma s = GammaAux...`? — actually Mathlib has `Complex.GammaAux` (the Hadamard-style construction!): `GammaAux n s := ...` — YES — Mathlib's Complex.Gamma is defined as the limit of GammaAux n s = n^s · n!/(s(s+1)···(s+n)) — the Euler/Gauss product form! That's a Weierstrass-ADJACENT definition. The log of the Euler product: log GammaAux involves Σ_m log(1 + s/m)-type terms — from which the digamma series could be derived: ψ(s) = lim [log n − Σ_{m=0}^n 1/(s+m)] (the Gauss form!) — ψ(s) = lim_{n→∞} (log n − Σ_{m=0}^n 1/(s+m)) — this is EXACTLY derivable from the GammaAux definition + logDeriv of the finite product + limit interchange. From the Gauss form, the Taylor coefficients at s = 1/2: ψ^{(j)}(1/2) = lim_n Σ_{m=0}^n (−1)^{j+1} j! (1/2 + m)^{−(j+1)} = (−1)^{j+1} j! Σ_{m≥0} (m + 1/2)^{−(j+1)} ✓ (termwise deriv of the uniform-on-compacts convergent series Σ 1/(s+m)... wait the Gauss form has the log n divergence-canceling — the differentiated series Σ 1/(s+m)^{j+1} for j ≥ 1 converges absolutely ✓ and the j = 0 derivative needs the log n − Σ 1/(s+m) with cancellation ✓ fine).\n\nSo (III) via the Gauss limit form: ψ^{(j)}(1/2) = (−1)^{j+1} j! Σ_{m≥0}(m+1/2)^{−(j+1)} for j ≥ 1, and Σ_{m≥0}(m+1/2)^{−(j+1)} = 2^{j+1} Σ (2m+1)^{−(j+1)} = 2^{j+1}·(odd-sum from m=0: 2m+1 = 1,3,5,...) = 2^{j+1}·lagariasA(j+1) ✓ DIRECTLY the odd-m encoding ✓.\n\nThe Lean work for (III):\n- ΓR′/ΓR(1+s) = −(log π)/2 + (1/2)ψ((1+s)/2) — logDeriv of Gammaℝ: Mathlib `Complex.Gammaℝ`? — wait — Complex.Gammaℝ s := π^{−s/2}·Γ(s/2) — Mathlib has `Complex.Gammaℝ`? — used in Main.lean: `Complex.Gammaℝ` ✓ exists. Its logDeriv: logDeriv Gammaℝ s = logDeriv (π^{−s/2}) s + (1/2)·logDeriv Γ (s/2) — via logDeriv_mul/comp — the π^{−s/2} piece: −(log π)/2·π^{−s/2}/π^{−s/2} = −(log π)/2 ✓. So Γ′R/ΓR(1+s) = −(log π)/2 + (1/2)ψ((1+s)/2) ✓ derivable.\n- The Taylor coefficients of ψ((1+s)/2) at s = 0: (1/2)ψ^{(j)}(1/2)·(1/2)^j/j!·... — coefficient of s^j in (1/2)ψ((1+s)/2): (1/2)·ψ^{(j)}(1/2)/j!·(1/2)^j = ψ^{(j)}(1/2)/(j!·2^{j+1}) — wait: ψ((1+s)/2) = ψ(1/2 + s/2) — Taylor in s: Σ_j ψ^{(j)}(1/2)/j!·(s/2)^j = Σ_j ψ^{(j)}(1/2)·2^{−j}/j!·s^j — then times 1/2: coefficient τ_j = ψ^{(j)}(1/2)·2^{−(j+1)}/j!. With ψ^{(j)}(1/2) = (−1)^{j+1}j!·2^{j+1}·lagariasA(j+1): τ_j = (−1)^{j+1}·lagariasA(j+1) ✓✓ for j ≥ 1. For j = 0: τ_0 = −(log π)/2 + (1/2)ψ(1/2) = −(log π)/2 + (1/2)(−2log2 − γ) = −(log π + 2log2 + γ)/2 = −(log 4π + γ)/2 = −a₁ ✓.\n\nSo (III) reduces to: **ψ^{(j)}(1/2) = (−1)^{j+1} j! Σ_{m≥0} (m+1/2)^{−(j+1)} for j ≥ 1** + the odd-sum-to-lagariasA identification (Σ_{m≥0}(m+1/2)^{−(j+1)} = 2^{j+1}·lagariasA(j+1) — needs lagariasA(j+1) = Σ_{m odd ≥ 1} m^{−(j+1)} = my definition ✓ + reindex m ↦ 2m+1 bijection between {m ≥ 0} and {odd ≥ 1} — a tsum reindexing via an Equiv (ℕ ≃ {odd naturals}) — `Equiv.tsum_eq` with the equiv m ↦ 2m+1... the map ℕ → {m // Odd m}: m ↦ 2m+1 is a bijection (Odd m ⟺ m = 2k+1 ✓ — `Nat.odd_iff` — the equiv needs building: Equiv.ofBijective) — and then the {odd} vs {odd ∧ ≥ 3} subtypes: my lagariasA = 1 + Σ_{odd ≥ 3} — and Σ_{m // Odd m} (1/(m+1/2)... hmm the Σ_{m≥0} (2m+1)^{−(j+1)} = Σ_{m odd ≥ 1} m^{−(j+1)} = 1 + Σ_{m odd ≥ 3} = lagariasA ✓ by def ✓ (the m=1 term = 1^{−(j+1)} = 1 ✓). So: Σ_{m≥0}(2m+1)^{−(j+1)} = lagariasA(j+1) — via the tsum over {m // Odd m} = the subtype {m // Odd m ∧ 1 ≤ m}... — my lagariasA uses {m // Odd m ∧ 3 ≤ m} with the +1 outside. The reindexing target: Σ_{m≥0}(2m+1)^{−(j+1)} — over ℕ — to lagariasA: 1 + Σ_{m : {odd ∧ ≥3}} m^{−(j+1)} — steps: Σ_{m≥0} g(2m+1) = Σ_{m ∈ {m // Odd m}} g(m) [equiv] = g(1) + Σ_{m ∈ {odd ∧ ≥3}} g(m) [omit m=1] = 1 + tail ✓. The omit step — {m // Odd m} = {⟨1, odd_one⟩} ∪ {m // Odd m ∧ 3 ≤ m}-ish — tsum_eq_add_tsum_omit (the name I couldn't find earlier — hmm — let me use: Summable tsum over {m // Odd m} = g(1) + rest — via `tsum_eq_add_tsum_ite`? or the union split {odd} = {1} ∪ {odd ∧ 3 ≤ m} as SETS — `Summable.tsum_set_union`? — there might be `tsum_union` for disjoint sets with finitely-supported... For subtypes: {m // Odd m} ≃ {m // m = 1} ⊕ {m // Odd m ∧ 3 ≤ m}? A Sum decomposition: `Equiv.sumCompl`? There's `Equiv.Set...`? Hmm — `tsum_eq_add_tsum_omit` — let me check its existence properly this time — search for \"omit\":\n\nActually — Mathlib: `tsum_eq_add_tsum_omit (hf : Summable f) (i : α) : ∑' x, f x = f i + ∑' x : {x // x ≠ i}, f x`? I have a strong feeling this exists (it's the standard \"peel one term\"). Earlier grep didn't find \"tsum_eq_add_tsum_omit\" but found \"tsum_eq_add_tsum_ite\". Let me check that name: `tsum_eq_add_tsum_ite (hf : Summable f) (b : β) : ∑' n, f n = f b + ∑' n, ite (n = b) 0 (f n)`? — ite-form ✓ usable too: then the ite-sum over {m // Odd m} equals the sum over {m // Odd m ∧ m ≠ 1} (via tsum_congr on the subtype... hmm the ite form is over the SAME index type: ∑' x : {odd}, if x = ⟨1,_⟩ then 0 else g x — and I want ∑' x : {odd ∧ 3 ≤ x}, g x — bridge via the equiv {x : {odd} // x ≠ ⟨1⟩} ≃ {m // Odd m ∧ 3 ≤ m} and `tsum_subtype`/`tsum_eq_of_...` — meh. Alternative — SKIP the {odd} middleman entirely: relate Σ_{m≥0}(2m+1)^{−(j+1)} DIRECTLY to my lagariasA's tail: {2m+1 : m ≥ 0} as m ranges ℕ gives 1, 3, 5, ... — and lagariasA's tail subtype is {m // Odd m ∧ 3 ≤ m}: the map m ↦ 2m+3 : ℕ → {odd ∧ ≥3}? bijection (odd ∧ ≥3 ⟺ m = 2k+3 ✓). And Σ_{m≥0} (2m+1)^{−(j+1)} = (2·0+1)^{−(j+1)} + Σ_{m≥0} (2(m+1)+1)^{−(j+1)} = 1 + Σ_{m≥0}(2m+3)^{−(j+1)} [tsum_eq_add_tsum... the peel of a ℕ-series: `tsum_eq_zero_add`? there's `tsum_eq_zero_add (hf : Summable f) : ∑' n, f n = f 0 + ∑' n, f (n+1)` ✓ `tsum_eq_zero_add` exists I'm sure]. Then Σ_{m≥0} g(2m+3) = Σ_{m : {odd ∧ ≥3}} g(m) via the equiv m ↦ 2m+3 (Equiv.tsum_eq ✓ exists — used in JacobiTheta). The equiv: (fun m => 2m+3) : ℕ ≃ {m // Odd m ∧ 3 ≤ m} — build via Equiv.ofBijective: injective ✓ (2m+3 = 2m'+3 ⟹ m = m' ✓ omega-ish); surjective: odd ∧ 3 ≤ m ⟹ m = 2k+3: m odd ⟹ m = 2k+1 (Nat.odd_iff); 3 ≤ 2k+1 ⟹ k ≥ 1; m = 2(k−1)+3 ✓. Fine — small bespoke equiv.\n\nSo (III) is: (III-a) ψ^{(j)}(1/2) formula for j ≥ 1 via the Gauss limit + Weierstrass-termwise-differentiation; (III-b) the odd-series reindexing; (III-c) j = 0 via digamma_one_half ✓ in Mathlib.\n\nThe heavy part of (III-a): deriving ψ^{(j)}(1/2) = (−1)^{j+1} j! Σ_{m≥0} (m+1/2)^{−(j+1)}. From Mathlib's Gamma definition: need `Complex.Gamma` as limit of GammaAux — check what exists: `Complex.Gamma_eq_GammaAux`? `tendsto_GammaAux`? The derivative route: ψ = logDeriv Gamma — ψ^{(j)}(1/2) = (d/ds)^j [Γ′/Γ](1/2) — hmm — computing j-th derivatives of logDeriv Gamma via the GammaAux limit: logDeriv GammaAux_n(s) = log n − Σ_{m=0}^n 1/(s+m) (each factor's logDeriv: n^s gives log n; s(s+1)...(s+n) gives Σ 1/(s+m)) — then ψ = lim of those (locally uniformly? — the logDeriv of a locally uniform limit ≠ limit of logDerivs in general — BUT Mathlib's LocallyUniformLimit.lean HAS `TendstoLocallyUniformlyOn.deriv` (line 22: \"Locally uniform convergence implies locally uniform [convergence of derivatives]\") ✓✓ so IF GammaAux_n → Gamma locally uniformly AND the logDerivs... careful: logDeriv Γ_n → logDeriv Γ needs: Γ_n → Γ locally uniformly (for the deriv convergence) AND Γ_n, Γ nonvanishing (for the quotient). The GammaAux convergence: does Mathlib state it? Let me check Mathlib/Analysis/SpecialFunctions/Gamma/Basic.lean for GammaAux and its convergence.\n\nActually — hmm, wait. Let me check whether Mathlib already has the digamma SERIES somewhere (maybe in a different file — the \"Gauss digamma theorem\" or the series for ψ):","signature":""},{"type":"text","text":"Checking Mathlib's Gamma definition (GammaAux) and any existing digamma series/derivative infrastructure."},{"type":"tool_use","id":"Bash_172","name":"Bash","input":{"command":"cd \"D:/Python/Quantum/zeta-23-lean/.lake/packages/mathlib\" && grep -rn \"GammaAux\\|GammaSeq\" Mathlib/Analysis/SpecialFunctions/Gamma/Basic.lean | head -12; echo ===; grep -rn \"theorem\\|lemma\" Mathlib/Analysis/SpecialFunctions/Gamma/Deriv.lean | grep -i \"digamma\\|logDeriv\\|iteratedDeriv\" | head -10","description":"Check GammaAux and Gamma derivative lemmas"},"extra_content":null}]