[{"type":"thinking","thinking":"I see the pattern. For the ContDiffAt of `logDeriv Gammaℝ (1+s)` at 0, I can build AnalyticAt directly:\n\nAnalyticAt ℂ (fun s => logDeriv Complex.Gammaℝ (1+s)) 0:\n- AnalyticAt Gammaℝ at 1: need it. differentiableAt_Gammaℝ_of_not_neg_nat gives DifferentiableAt; for AnalyticAt: Gammaℝ is analytic on its domain. Is there `analyticAt_Gammaℝ`? Hmm — in LiFEntire or LiCircleBound, `differentiableAt_Gammaℝ_of_not_neg_nat` was used. For AnalyticAt: Gammaℝ = π^{−s/2}·Γ(s/2) — analytic where Γ(s/2) analytic (s/2 ∉ −ℕ) — there might be `Complex.analyticAt_Gamma` (Gamma analytic off −ℕ: `Complex.Gamma_analyticAt`?). Simpler: logDeriv Gammaℝ (1+s) =ᶠ[𝓝 0] (−(log π)/2 + (1/2)·digamma((1+s)/2)) (logDeriv_Gammaℝ_one_add, used above) and the RHS is analytic at 0 (digamma analytic at 1/2, comp). AnalyticAt of a function eventually-equal to an analytic one: `EventuallyEq.analyticAt`? Hmm — ContDiffAt under EventuallyEq: `ContDiffAt.congr_of_eventuallyEq`? Hmm.\n\nActually — for iteratedDeriv_add I need ContDiffAt of each summand. I can avoid needing ContDiffAt of logDeriv Gammaℝ (1+s) directly by structuring differently: I don't need to split MY eventual RHS into pieces with ContDiffAt of the ORIGINAL functions — I need ContDiffAt of the three RHS pieces:\n1. fun s => (1+s)⁻¹ — analytic at 0 ✓ easy.\n2. hLiExt — analytic at 0 ✓ (hLiExt_analyticAt_zero).\n3. fun s => logDeriv Gammaℝ (1+s) — use the eventuallyEq with the digamma form: ContDiffAt.congr_of_eventuallyEq? There IS `ContDiffAt.congr_of_eventuallyEq`? Hmm — actually iteratedDeriv_congr_of_eventuallyEq handles the derivative value; but iteratedDeriv_add needs ContDiffAt hypotheses for the pieces as written. I can build ContDiffAt of piece 3 via the eventuallyEq: if f =ᶠ[𝓝 x] g and g is ContDiffAt, then f is ContDiffAt: `ContDiffAt.congr_of_eventuallyEq`? Let me check: there is `ContDiffAt.congr_of_eventuallyEq (h : f₁ =ᶠ[𝓝 x] f) : ContDiffAt 𝕜 n f x → ContDiffAt 𝕜 n f₁ x`? I believe `ContDiffAt.congr_of_eventuallyEq` exists in Mathlib. And the digamma form is ContDiffAt via the hcd0/hcd pattern above.\n\nAlternatively directly AnalyticAt: `Complex.Gammaℝ` analyticAt at non-integer... Let me check for an existing lemma: grep \"analyticAt_Gammaℝ\" or \"Gammaℝ.*Analytic\" in the repo.\n\nActually — LiFEntire line ~273: `have hB : AnalyticAt ℂ seriesFun 1 := ...`. For Gammaℝ: LiCircleBound has `differentiableAt_Gammaℝ_of_not_neg_nat`. For analytic: in LiDigamma there's `logDeriv_Gammaℝ_one_add` (the eventuallyEq used above) — the cleanest is: piece 3 ContDiffAt via congr_of_eventuallyEq with the digamma form whose ContDiffAt is hcd (built exactly as above lines 208-219).\n\nLet me check ContDiffAt.congr_of_eventuallyEq exists. Also check hLiExt_analyticAt_zero's exact name, and zetaA def/zetaA_apply, differentiableAt_riemannZeta, one_div form lemmas. Probe time. Also `AnalyticAt.comp_of_eq` (used above) exists ✓.\n\nAlso for the punctured identity I need:\n- zetaA def: `def zetaA`? LiResidue:96 uses zetaA in hLi. zetaA w := (w−1)·ζ(w)? Let me check.\n- zetaA_apply (used in LiMoments:331: `zetaA_apply hs1` gave zetaA s = (s−1)*riemannZeta s).\n- deriv of zetaA at 1+s: product rule.\n- logDeriv_Gammaℝ_one_add : statement? (used at LiArchimedean:200: logDeriv_Gammaℝ_one_add h1 h2 : logDeriv Gammaℝ (1+s) = −(log π)/2 + (1/2)·digamma((1+s)/2), with h1 : Gamma ((1+s)/2) ≠ 0, h2 : ∀ m, (1+s)/2 ≠ −m).\n\nFor the punctured gFun identity at 1+s, mirror gFun_eq_momentB's structure:\n- hexp : xiL =ᶠ[𝓝 (1+s)] product form (xiL_eq_mul; needs 1+s ≠ 0 and 1+s ≠ 1: s ≠ −1 and s ≠ 0 ✓ eventually).\n- hΛeq : completedRiemannZeta =ᶠ[𝓝 (1+s)] ΓR·ζ (needs ΓR(1+s) ≠ 0 eventually + 1+s ≠ 0 ✓).\n- logDeriv_mul splits with nonvanishing at 1+s: (1+s)·s/2 ≠ 0 (s ≠ 0, s ≠ −1), Λ(1+s) ≠ 0 (since ζ(1+s) ≠ 0 via completedRiemannZeta_eq_zero_iff), ΓR(1+s) ≠ 0 (ΓR(1) = 1, eventually), ζ(1+s) ≠ 0 (eventually: ζ has a pole at 1 → |ζ| → ∞ → nonzero nearby: zetaA (1+s) ≠ 0 eventually (zetaA_ne_zero_near_one used in good_eventually_one) ⟹ ζ(1+s) ≠ 0 for s ≠ 0 since zetaA (1+s) = s·ζ(1+s)).\n\nThen:\ngFun (1+s) = logDeriv xiL (1+s) [by gFun def — actually gFun (1+s) IS logDeriv xiL (1+s) by def of gFun: gFun := fun s => logDeriv xiL s, so gFun (1+s) = logDeriv xiL (1+s) rfl ✓ no rewrite needed]\n= logDeriv (fun t => (t(t−1)/2)·completedRiemannZeta t) (1+s) [hexp deriv/point congruence]\n= logDeriv (fun t => t(t−1)/2) (1+s) + logDeriv completedRiemannZeta (1+s) [logDeriv_mul]\n= [1/(1+s) + 1/s] + [logDeriv Gammaℝ (1+s) + logDeriv riemannZeta (1+s)] [hPlog at 1+s; hlogΛ via hΛeq + logDeriv_mul]\n✓ then combine with hLiExt s = 1/s + logDeriv riemannZeta (1+s).\n\nhLiExt identity: hLiExt s = (1 − hLi (1+s))/s (hLiExt_of_ne hs). hLi (1+s) = 1 − s·(deriv zetaA (1+s)/zetaA (1+s)) [hLi def: hLi w = 1 − (w−1)·(deriv zetaA w/zetaA w) at w = 1+s: 1 − s·(...)].\n1 − hLi (1+s) = s·(deriv zetaA (1+s)/zetaA (1+s)).\n(1 − hLi (1+s))/s = deriv zetaA (1+s)/zetaA (1+s) [field_simp s ≠ 0... mul_div_cancel].\n\nderiv zetaA (1+s): zetaA w = (w−1)·ζ(w)? check zetaA def. If zetaA w := (w−1)*riemannZeta w: deriv zetaA (1+s) = deriv ((fun w => (w−1)) * ζ) (1+s) = 1·ζ(1+s) + s·deriv ζ (1+s) [deriv_mul: differentiable]. \n\nHmm wait — I need deriv zetaA at the point (1+s), not deriv of (fun s => zetaA (1+s)). deriv_mul gives deriv (f*g) (1+s) = deriv f (1+s)·g(1+s) + f(1+s)·deriv g (1+s) with f = fun w => w−1, g = ζ ✓.\n\nderiv zetaA (1+s)/zetaA (1+s) = [ζ(1+s) + s·ζ′(1+s)]/(s·ζ(1+s)) = 1/s + ζ′(1+s)/ζ(1+s) = 1/s + logDeriv riemannZeta (1+s) [field_simp with s ≠ 0, ζ(1+s) ≠ 0].\n\nCheck zetaA's def and zetaA_apply.","signature":""},{"type":"text","text":"Checking zetaA's definition and a few remaining signatures."},{"type":"tool_use","id":"Grep_99","name":"Grep","input":{"pattern":"def zetaA|lemma zetaA_apply|theorem zetaA_apply|lemma logDeriv_Gammaℝ_one_add|hLiExt_analyticAt_zero|zetaA_ne_zero_near_one","path":"D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1","output_mode":"content","-n":true,"-A":6,"head_limit":60},"extra_content":null}]