[{"type":"thinking","thinking":"`Complex.analyticAt_cexp : AnalyticAt ℂ exp z` (ExpDeriv.lean:52, inside Complex namespace presumably) and `AnalyticAt.cexp (fa : AnalyticAt ℂ f x) : AnalyticAt ℂ (exp ∘ f) x` ✓ — even better, use `.cexp` directly.\n\nNow — the cpow piece: (fun s => (π:ℂ)^(−s/2)) = (fun s => exp (log π * (−s/2))): cpow_def_of_ne_zero — check its exact form: `Complex.cpow_def_of_ne_zero (hx : x ≠ 0) : x ^ y = exp (log x * y)`. Then AnalyticAt of s ↦ log π * (−s/2): linear: (analyticAt_const.mul analyticAt_id) after showing = fun s => (log π * (−1/2)) * s? Or: (fun s => Complex.log π * (−s/2)): = (fun s => (Complex.log π * (-1/2)) * s): funext ring. Then `AnalyticAt.cexp` of that.\n\nLet me write:\n```lean\nhave hcpi : AnalyticAt ℂ (fun s => (Real.pi : ℂ) ^ (-s / 2)) (-2) := by\n  have heq : (fun s => (Real.pi : ℂ) ^ (-s / 2))\n      = fun s => Complex.exp ((Complex.log (Real.pi : ℂ) * (-1 / 2)) * s) := by\n    funext s\n    rw [Complex.cpow_def_of_ne_zero (by norm_num : (Real.pi : ℂ) ≠ 0)]\n    ring_nf\n  rw [heq]\n  exact (analyticAt_const.mul analyticAt_id).cexp\n```\ncpow_def_of_ne_zero: π^(−s/2) = exp (log π * (−s/2)) — goal after funext: π^(−s/2) = exp((log π·(−1/2))·s): rw [cpow_def_of_ne_zero] : LHS → exp (log π * (−s/2)): goal exp(log π·(−s/2)) = exp((log π·(−1/2))·s): congr 1; ring ✓ (ring_nf might handle the congr automatically? exp args equality: ring_nf should close `exp a = exp b` when a = b by ring_nf? ring_nf normalizes inside atoms ✓ ring_nf closes if log π·(−s/2) and (log π·(−1/2))·s have same normal form ✓). So: rw [cpow_def_of_ne_zero ...]; ring_nf ✓.\n\n`analyticAt_const.mul analyticAt_id : AnalyticAt ℂ (fun s => c * s)?` — analyticAt_const : AnalyticAt 𝕜 (fun _ => v) x; .mul analyticAt_id: AnalyticAt (fun s => c * id s) = fun s => c*s ✓ matches after heq's form `(log π * (−1/2)) * s` ✓. Then `.cexp` : AnalyticAt (exp ∘ f) — but heq's RHS is fun s => exp (c*s) = exp ∘ (fun s => c*s) ✓ defeq? `AnalyticAt.cexp (fa) : AnalyticAt ℂ (exp ∘ f) x` — our goal after rw [heq]: AnalyticAt ℂ (fun s => Complex.exp ((...) * s)) (−2) — is `fun s => exp (c*s)` defeq to `exp ∘ fun s => c*s`? Yes (∘ unfolds) — exact might accept; if not, `exact (analyticAt_const.mul analyticAt_id).cexp` with unification should work.\n\nGamma piece:\n```lean\nhave hGamma : AnalyticAt ℂ (fun s => Complex.Gamma (s / 2 + 2)) (-2) := by\n  have hd : DifferentiableOn ℂ (fun s => Complex.Gamma (s / 2 + 2)) (Metric.ball (-2) 1) := by\n    intro s hs\n    have hg : DifferentiableAt ℂ Complex.Gamma (s / 2 + 2) := by\n      apply Complex.differentiableAt_Gamma\n      intro m hm  -- hm : s/2+2 = −↑m\n      -- contradiction with hs : s ∈ ball (−2) 1\n      ...\n    exact hg.comp s (by fun_prop)  -- fun s => s/2+2 differentiable\n  exact ((hd.analyticOnNhd Metric.isOpen_ball) (-2) (Metric.mem_ball_self zero_lt_one))\n```\nDifferentiableOn.analyticOnNhd gives AnalyticOnNhd ℂ f (ball ...) — then apply at point: AnalyticOnNhd ℂ f s is ∀ x ∈ s, AnalyticAt f x — so `(hd.analyticOnNhd Metric.isOpen_ball) (-2) (Metric.mem_ball_self zero_lt_one)` ✓.\n\nThe m-contradiction: hm : s/2 + 2 = -(m:ℂ):\n```lean\n      have hs_eq : s = -2 * (m : ℂ) - 4 := by linear_combination 2 * hm\n```\nCheck: 2*(s/2+2 − (−m)) = s + 4 + 2m; goal s − (−2m−4) = s + 2m + 4 ✓ equal ✓ linear_combination 2*hm works.\n```lean\n      rw [hs_eq, Metric.mem_ball, dist_eq_norm] at hs\n      -- hs : ‖−2m−4 − (−2)‖ < 1: = ‖−(2m+2)‖ = 2m+2 ≥ 2\n      have h1 : ((-2 : ℂ) * (m : ℂ) - 4) - -2 = -(((2:ℝ) * m + 2 : ℝ) : ℂ) := by push_cast; ring\n      rw [h1, norm_neg, Complex.norm_real, Real.norm_eq_abs] at hs\n      rw [abs_of_nonneg (by positivity : (0:ℝ) ≤ 2 * (m:ℝ) + 2)] at hs\n      -- hs : 2m + 2 < 1 — but 2m+2 ≥ 2\n      have : (2:ℝ) ≤ 2 * (m:ℝ) + 2 := by have hm0 : (0:ℝ) ≤ m := by positivity; linarith\n      linarith\n```\nWait — norm_neg : ‖−a‖ = ‖a‖ ✓; Complex.norm_real : ‖(r:ℂ)‖ = ‖r‖ (real norm) then Real.norm_eq_abs : ‖r‖ = |r| ✓. hs : |2m+2| < 1 → 2m+2 < 1 — contradiction with ≥ 2 ✓ linarith.\n\nHmm — h1's ring: (−2)·↑m − 4 − (−2) = −2m − 2 = −(2m+2) ✓ push_cast; ring ✓. But careful the coercion: ((2:ℝ)*m + 2 : ℝ) : ℂ — push_cast on the LHS ring normal form... The claim is an equality in ℂ: LHS has ↑(m:ℕ); RHS ↑((2:ℝ)*↑m+2): push_cast normalizes casts ✓ ring ✓.\n\n`hg.comp s (by fun_prop)` — DifferentiableAt.comp (x : 𝕜) (hf : DifferentiableAt 𝕜 f (g x)) (hg : DifferentiableAt 𝕜 g x) : DifferentiableAt 𝕜 (f ∘ g) x — gives (f ∘ g) — goal is fun s => Gamma (s/2+2) — (Gamma ∘ fun s => s/2+2) defeq ✓. fun_prop proves DifferentiableAt ℂ (fun s => s/2+2) s ✓.\n\nhΛ_an:\n```lean\nhave hΛ_an : AnalyticAt ℂ completedRiemannZeta (-2) := by\n  have hd : DifferentiableOn ℂ completedRiemannZeta (Metric.ball (-2) 1) := by\n    intro s hs\n    have hs0 : s ≠ 0 := by\n      rintro h\n      rw [h, Metric.mem_ball, dist_eq_norm] at hs\n      -- ‖0 − (−2)‖ = 2 < 1 contradiction\n      norm_num at hs  -- ‖(0:ℂ) − −2‖ = ‖2‖ = 2: norm_num with norm... simp [sub_neg_eq_add]... \n    have hs1 : s ≠ 1 := by\n      rintro h\n      rw [h, Metric.mem_ball, dist_eq_norm] at hs\n      norm_num at hs  -- ‖1 − (−2)‖ = 3 < 1 ✗\n    exact differentiableAt_completedZeta hs0 hs1\n  exact (hd.analyticOnNhd Metric.isOpen_ball) (-2) (Metric.mem_ball_self zero_lt_one)\n```\nnorm_num at hs for ‖(0:ℂ) − −2‖ < 1: (0) − −2 = 2: ‖(2:ℂ)‖ = 2: simp [sub_neg_eq_add, Complex.norm_ofNat]? `Complex.norm_ofNat`? there's `Complex.norm_two`? Let me use: rw [show ((0:ℂ) - -2) = 2 by norm_num] then `Complex.norm_ofNat (2)`? Hmm — norm of numeral in ℂ: `Complex.norm_ofNat` exists? There's `Complex.norm_ofNat (n : ℕ) [n.AtLeastTwo] : ‖(ofNat(n) : ℂ)‖ = ofNat(n)`? Probably. Or via Complex.norm_real after identifying (2:ℂ) = ((2:ℝ):ℂ): simp [Complex.norm_real]? To be safe:\n```lean\n      rw [h, Metric.mem_ball, dist_eq_norm] at hs\n      have : ((0 : ℂ) - -2) = ((2 : ℝ) : ℂ) := by norm_num\n      rw [this, Complex.norm_real, Real.norm_eq_abs] at hs\n      norm_num at hs  -- |2| < 1 false\n```\nFor s = 1: ((1:ℂ) − −2) = ((3:ℝ):ℂ) ✓ similar.\n\nThe eventuallyEq hEv — full assembly:\n```lean\nhave hEv : ∀ᶠ s in 𝓝 (-2 : ℂ),\n    riemannZeta s = (s - -2) ^ 1 • (completedRiemannZeta s / Hfun s) := by\n  have hHne_ev : ∀ᶠ s in 𝓝 (-2 : ℂ), Hfun s ≠ 0 :=\n    hHfun_an.continuousAt.eventually_ne hHfun_ne\n  filter_upwards [Metric.ball_mem_nhds (-2) (zero_lt_one : (0:ℝ) < 1), hHne_ev]\n    with s hs1 hHne\n  by_cases hs : s = -2\n  · subst hs\n    have hz0 : riemannZeta (-2) = 0 := by\n      have h := riemannZeta_neg_two_mul_nat_add_one 0\n      norm_num at h  -- h : riemannZeta (-2) = 0? −2*(0+1) = −2·1 = −2 ✓ norm_num\n      exact h\n    rw [hz0]\n    simp  -- (−2 − −2)^1 • (…) = 0 • … = 0\n  · have hs0 : s ≠ 0 := by ... (ball excludes 0, same as above)\n    have hΓR : Complex.Gammaℝ s ≠ 0 := ... (the m-argument)\n    have hΛ := completedRiemannZeta_eq_Gammaℝ_mul hs0 hΓR\n    have hH : (s + 2) * Complex.Gammaℝ s = Hfun s := Hfun_eq s hs hs0\n    rw [pow_one, smul_eq_mul]\n    -- goal: riemannZeta s = (s - -2) * (Λ s / Hfun s)\n    rw [show s - -2 = s + 2 by ring]\n    -- ζ = Λ/ΓR:\n    have hz : riemannZeta s = completedRiemannZeta s / Complex.Gammaℝ s := by\n      rw [hΛ]; field_simp [hΓR]  -- Λ = ΓR·ζ ⟹ ζ = Λ/ΓR: mul_div_cancel? (ΓR·ζ)/ΓR = ζ: mul_div_cancel_left₀\n    rw [hz]\n    -- Λ/ΓR = (s+2)·(Λ/Hfun) ⟺ cross: Λ·Hfun = (s+2)·Λ·ΓR\n    rw [div_mul_eq_mul_div, div_eq_div_iff hΓR hHne]  -- Λ·Hfun = (s+2)·Λ·ΓR\n    -- via hH: Hfun = (s+2)·ΓR\n    rw [← hH]; ring\n```\nHmm direction check on div_eq_div_iff: `div_eq_div_iff (hb : b ≠ 0) (hd : d ≠ 0) : a / b = c / d ↔ a * d = c * b` — earlier the error told us div_eq_div_iff takes ≠ 0 args ✓. Goal after rw [hz]: Λ s / ΓR s = (s+2) * (Λ s / Hfun s). rw [div_mul_eq_mul_div]: RHS (s+2)·(Λ/Hfun) → ((s+2)·Λ)/Hfun. Goal: Λ/ΓR = ((s+2)·Λ)/Hfun. rw [div_eq_div_iff hΓR hHne]: Λ·Hfun = (s+2)·Λ·ΓR. rw [← hH]: hH : (s+2)·ΓR = Hfun; ← hH rewrites Hfun → (s+2)·ΓR: goal: Λ·((s+2)ΓR) = (s+2)·Λ·ΓR: ring ✓.\n\nhz: goal riemannZeta s = Λ s / ΓR s: rw [hΛ] — hΛ : Λ s = ΓR s * ζ s — rewriting goal's Λ s → ΓR s·ζ s: riemannZeta s = (ΓR s · riemannZeta s)/ΓR s: then `mul_div_cancel_left₀`: `mul_div_cancel_left₀ (a : G) (h : b ≠ 0)?` — signature: `mul_div_cancel_left₀ (a) (h : b ≠ 0) : b * a / b = a`? or (h : b ≠ 0) (a)? In current Mathlib: `mul_div_cancel_left₀ (hb : b ≠ 0) (a : G) : b * a / b = a`? I'll do: rw [hΛ, mul_div_cancel_left₀ _ hΓR]?? — adjust at compile: `(mul_div_cancel_left₀ _ hΓR).symm`? The exact form: mul_div_cancel_left₀ (a : α) (h : b ≠ 0) : b * a / b = a — older form had (a) first. If issues: field_simp [hΓR] then ring.\n\nAt s = −2 case: goal after rw [hz0]: (−2 − −2)^1 • (Λ(−2)/Hfun(−2)) = ... wait the goal is `riemannZeta (−2) = (−2 − −2)^1 • G(−2)` and after rw [hz0] (LHS → 0): `0 = (−2 − −2)^1 • G(−2)`: simp [pow_one, smul_eq_mul] : −2−−2 = 0, 0·_ = 0 ✓ simp closes (sub_self, zero_mul).\n\nhΓR proof:\n```lean\n    have hΓR : Complex.Gammaℝ s ≠ 0 := by\n      rw [Complex.Gammaℝ_eq_zero_iff]  -- need exact name/namespace: Complex.Gammaℝ_eq_zero_iff\n      push Not\n      rintro ⟨m, hm⟩  -- hm : s = -(2 * ↑m)\n      rcases Nat.eq_zero_or_pos m with hm0 | hmpos\n      · subst hm0\n        norm_num at hm  -- hm : s = 0\n        exact hs0 hm\n      · rcases eq_or_ne m 1 with hm1 | hm1\n        · subst hm1\n          norm_num at hm  -- hm : s = −2\n          exact hs hm\n        · have hm2 : 2 ≤ m := by omega\n          rw [hm, Metric.mem_ball, dist_eq_norm] at hs1\n          have h1 : (-(2 * (m : ℂ))) - -2 = ((2 - 2 * (m : ℝ) : ℝ) : ℂ) := by push_cast; ring\n          rw [h1, Complex.norm_real, Real.norm_eq_abs] at hs1\n          have hle : (2:ℝ) - 2 * m ≤ 0 := by have : (2:ℝ) ≤ m := by exact_mod_cast hm2; linarith\n          rw [abs_of_nonpos hle] at hs1\n          -- hs1 : -(2 − 2m) < 1 i.e. 2m − 2 < 1\n          have hmR : (2:ℝ) ≤ (m : ℝ) := by exact_mod_cast hm2\n          nlinarith [hs1, hmR]  -- 2m−2 < 1 with m ≥ 2: 2m−2 ≥ 2 ⟹ contradiction\n```\nnlinarith: hs1 : −(2−2m) < 1 ⟹ 2m − 2 < 1; hmR : 2 ≤ m ⟹ 2m−2 ≥ 2 > 1 ✓ linarith should do (linear!). Use linarith [hs1, hmR] — but hs1 has the form `-(2 − 2 * ↑m) < 1` ✓ linear ✓.\n\nWait — push Not on `¬ ∃ n, s = −(2*↑n)` then rintro ⟨m, hm⟩: push Not transforms ¬∃ into ∀¬... then rintro doesn't apply. Wrong order: the goal after rw is `¬ ∃ n : ℕ, s = -(2 * n)`. Just `rintro ⟨m, hm⟩` directly (¬P = P → False) ✓. No push Not needed.\n\nAlso `Complex.Gammaℝ_eq_zero_iff` — is Gammaℝ under Complex namespace? Deligne.lean — check namespace. The lemma `Gammaℝ_eq_zero_iff` from my earlier sed output: line 74 `lemma Gammaℝ_eq_zero_iff {s : ℂ} : Gammaℝ s = 0 ↔ ∃ n : ℕ, s = -(2 * n)` — namespace in Deligne.lean: likely `namespace Complex`. With `open Complex` it's fine either way; I'll write `Complex.Gammaℝ_eq_zero_iff` and adjust.\n\nThe Hfun_eq (e): one more check on Gamma_add_one signature: `Complex.Gamma_add_one (s : ℂ) (h2 : s ≠ 0) : Gamma (s + 1) = s * Gamma s` ✓ (Basic.lean:311). \n\nhz's rewrite: careful — rw [hΛ] rewrites `completedRiemannZeta s` in the goal — goal is `riemannZeta s = completedRiemannZeta s / Complex.Gammaℝ s` — rw [hΛ] : `riemannZeta s = Gammaℝ s * riemannZeta s / Gammaℝ s` — then mul_div_cancel_left₀: b * a / b = a with b = Gammaℝ s ✓ `mul_div_cancel_left₀ (riemannZeta s) hΓR`? The exact argument order — check at compile.\n\nzeroMult_neg_two final assembly:\n```lean\ntheorem zeroMult_neg_two : zeroMult (-2) = 1 := by\n  have hHfun_an : ... \n  ...\n  have horder : analyticOrderAt riemannZeta (-2) = 1 :=\n    (hzeta_an.analyticOrderAt_eq_natCast).mpr\n      ⟨fun s => completedRiemannZeta s / Hfun s, hG_an, hG_ne, hEv⟩\n  rw [zeroMult, horder]\n  rfl  -- (1 : ℕ∞).toNat = 1\n```\nHmm — `(analyticOrderAt riemannZeta (-2)).toNat` with analyticOrderAt = (1 : ℕ∞): toNat of ℕ∞: is ℕ∞ = WithTop ℕ in this Mathlib? `.toNat` on WithTop ℕ: `WithTop.toNat`? exists? Hmm — the Zeta23 def compiled, so .toNat exists on whatever analyticOrderAt returns. If analyticOrderAt : ℕ∞ and ℕ∞ := WithTop ℕ: WithTop.toNat?? I don't think WithTop has toNat... Maybe ℕ∞ has its own toNat? There's `ENat`... In current Mathlib: `ℕ∞` is notation for `WithTop ℕ`? and `WithTop.toNat`? Hmm. Actually — I recall analyticOrderAt returns `ℕ∞` = PartENat?? no. Let me just check: grep \"def analyticOrderAt\" in Mathlib.\n\nAlso hzeta_an:\n```lean\n  have hG_an : AnalyticAt ℂ (fun s => completedRiemannZeta s / Hfun s) (-2) :=\n    hΛ_an.div hHfun_an hHfun_ne\n  have hRHS_an : AnalyticAt ℂ (fun s => (s - -2) ^ 1 • (completedRiemannZeta s / Hfun s)) (-2) := by\n    -- (s−−2)^1 • G s = (s+2)·G s: prove via congr from mul version\n    have : AnalyticAt ℂ (fun s => (s + 2) * (completedRiemannZeta s / Hfun s)) (-2) :=\n      ((analyticAt_id.add analyticAt_const)).mul hG_an\n    apply this.congr  -- f =ᶠ g then g analytic: congr (hf : AnalyticAt f) (hg : f =ᶠ g) : AnalyticAt g\n    filter_upwards with s\n    simp [pow_one, smul_eq_mul, sub_neg_eq_add]\n  have hzeta_an : AnalyticAt ℂ riemannZeta (-2) := hRHS_an.congr hEv\n```\nsmul on ℂ·ℂ: `(s−−2)^1 • G s` — • is ℂ acting on ℂ (mul). AnalyticAt.smul? For 𝕜-smooth: `AnalyticAt.smul (hf : AnalyticAt 𝕜 f x) (hg : AnalyticAt 𝕜 g x) : AnalyticAt (f • g)` for f 𝕜-valued, g E-valued — with E = ℂ, 𝕜 = ℂ ✓ so ((analyticAt_id.sub analyticAt_const).pow 1).smul hG_an directly:\n```lean\n  have hRHS_an : AnalyticAt ℂ (fun s => (s - -2) ^ 1 • (completedRiemannZeta s / Hfun s)) (-2) :=\n    (((analyticAt_id.sub analyticAt_const).pow 1).smul hG_an)\n```\nBut the function must match syntactically: fun s => ((s − −2)^1) • (Λ s / Hfun s) — the smul version: (fun s => (s−−2)^1) • (fun s => ...) — AnalyticAt.smul gives (f • g) x = f x • g x ✓ Pi.smul_apply defeq ✓ should unify. If not, congr route. Try smul first; fallback: .mul + congr with simp [pow_one, smul_eq_mul].\n\nHmm — analyticAt_id : AnalyticAt 𝕜 id z — but our function is fun s => s − −2 = fun s => s − (−2): analyticAt_id.sub analyticAt_const : AnalyticAt (fun s => id s − const) ✓ defeq to fun s => s − −2 ✓.\n\nLet me check analyticOrderAt's return type quickly, then write the whole §5 and iterate.","signature":""},{"type":"text","text":"Checking `analyticOrderAt`'s return type for the final `.toNat` step."},{"type":"tool_use","id":"Bash_108","name":"Bash","input":{"command":"cd D:/Python/Quantum/zeta-23-lean/.lake/packages/mathlib && grep -rn \"noncomputable def analyticOrderAt\\|def analyticOrderAt\" Mathlib/Analysis/Analytic/Order.lean | head -3 && grep -n \"ℕ∞\\|WithTop\" Mathlib/Analysis/Analytic/Order.lean | head -5","description":"Check analyticOrderAt return type"},"extra_content":null}]