[{"type":"thinking","thinking":"The summability: `ArithmeticFunction.LSeriesSummable_vonMangoldt (h : 1 < s.re)` + `summable_norm_iff`. ✓\n\nNow write piece 1. The steps:\n\n```lean\n/-- The term-norm form: ‖LSeries.term Λ 2 n‖ = if n = 0 then 0 else Λ n / n². -/\nlemma dirichletTerm_norm_eq (n : ℕ) :\n    ‖LSeries.term (fun n => (ArithmeticFunction.vonMangoldt n : ℂ)) (2 : ℂ) n‖\n      = if n = 0 then (0 : ℝ) else (ArithmeticFunction.vonMangoldt n) / (n : ℝ) ^ 2 := by\n  rcases eq_or_ne n 0 with rfl | hn\n  · simp [LSeries.term]\n  · rw [LSeries.term_def, if_neg hn]\n    -- ‖(Λ n : ℂ) / (n:ℂ)^(2:ℂ)‖ = Λ n / n²\n    rw [norm_div]\n    have h1 : ‖((ArithmeticFunction.vonMangoldt n : ℂ))‖ = ArithmeticFunction.vonMangoldt n := by\n      -- Λ n : ℂ is a nonneg real cast\n      rw [show (ArithmeticFunction.vonMangoldt n : ℂ)\n          = ((ArithmeticFunction.vonMangoldt n : ℝ) : ℂ) by push_cast; ring, Complex.norm_real]\n      exact abs_of_nonneg ArithmeticFunction.vonMangoldt_nonneg\n    have h2 : ‖((n : ℂ)) ^ (2 : ℂ)‖ = (n : ℝ) ^ 2 := by\n      have hn0 : (0 : ℝ) < n := by exact_mod_cast Nat.pos_of_ne_zero hn... \n      -- ‖(n:ℂ)^(2:ℂ)‖ = (n:ℝ)^((2:ℂ).re) = n^2\n      rw [Complex.norm_cpow_eq_rpow_re_of_pos ...] hmm\n```\nThe cpow norm: `Complex.norm_cpow_eq_rpow_re_of_pos (hx : 0 < x) (s : ℂ) : ‖(x : ℂ) ^ s‖ = x ^ s.re`? Let me check the exact name. In Mathlib: `Complex.norm_cpow_eq_rpow_re_of_pos (hx : 0 < x) : ‖(x : ℂ) ^ y‖ = x ^ y.re` — I think it's `Complex.norm_cpow_eq_rpow_re_of_pos`. Then ((2:ℂ).re = 2) and (n:ℝ)^(2:ℝ) = n² via Real.rpow_two? `Real.rpow_two (x) : x^(2:ℝ) = x^2`? there's rpow_two : x ^ (2:ℝ) = x * x. Then n*n = n² ring. OK. But hmm — is (n:ℂ)^(2:ℂ) a cpow — the term is f n / n^s with n : ℕ coerced and s : ℂ — n^s is `HPow ℂ ℂ` = cpow ✓.\n\nAlternatively avoid the cpow-norm: ‖(n:ℂ)^(2:ℂ)‖ = ‖(n:ℂ)‖^2 via... there's `Complex.norm_cpow_natCast`? hmm. Simplest might be: (n:ℂ)^(2:ℂ) = (n:ℂ)^2 (cpow with a natCast exponent: `cpow_two`? no — (x:ℂ)^((2:ℕ):ℂ) = x^2 via `cpow_natCast`? hmm: `Complex.cpow_natCast (x : ℂ) (n : ℕ) : x ^ (n : ℂ) = x ^ n`? I believe `cpow_natCast` exists (x ^ (↑n) = x ^ n). Then norm: ‖(n:ℂ)²‖ = ‖(n:ℂ)‖² = n² (norm_pow + Complex.norm_natCast). Let me use that: `Complex.cpow_natCast`? hmm the name could be `cpow_natCast` in root namespace. Check at build.\n\nThen:\n```lean\n/-- dirichletC2 = Σ' Λ(n)/n² (the nonzero terms). -/\nlemma dirichletC2_eq : dirichletC2\n    = ∑' n : ℕ, if n = 0 then (0 : ℝ) else ArithmeticFunction.vonMangoldt n / (n : ℝ) ^ 2 := by\n  rw [dirichletC2]\n  exact tsum_congr fun n => dirichletTerm_norm_eq n\n```\n\nThen:\n```lean\ntheorem dirichletC2_le_one : dirichletC2 ≤ 1 := by\n  rw [dirichletC2_eq]\n  -- majorant g n = if n = 0 then 0 else log n / n²\n  ...\n```\nPlan: \n- hsum: Summable (fun n => if n = 0 then 0 else Λ n/n²) — from the LSeries summability via the norm form (the dirichletTerm_norm_eq congr + hsum2-style).\n- hlog: Summable (fun n => if n = 0 then (0:ℝ) else Real.log n / n²) — via the integral test (antitone tail) or from... hmm, need summability for tsum_le_tsum. The log n/n² summability: `Summable` of n ↦ log n/n²: from the comparison with n^{−3/2}·(log n/√n → 0)... or directly via the integral test (AntitoneOn.summable_of_integrableOn_Ioi_zero needs Ici 0 antitone — log x/x² is NOT antitone on [0, e^{1/2}]). Alternative: summable_nat_add_iff: Σ_{n≥4} log n/n² summable + finite head. The tail's summability from the integral test (f antitone on Ici 3 ✓ + integrable on Ioi 3 ✓ from the FTC). So hlog := (summable_nat_add_iff 4).mpr (tail summable). The exact lemma: `summable_nat_add_iff (k : ℕ) : Summable (fun n => f (n + k)) ↔ Summable f` ✓ exists.\n- tsum_le_tsum pointwise (Λ n ≤ log n) then the log-series bound: Σ' logseries = Σ_{n<4} + Σ_{n≥4}: via sum_add_tsum_nat_add 4: ∑' n, f n = (∑ n ∈ range 4, f n) + ∑' n, f (n+4): head: f 0 = 0 (if-pos), f 1 = log 1/1 = 0, f 2 = log 2/4, f 3 = log 3/9. Tail: ∑' n, log(n+4)/(n+4)² ≤ ∫ Ioi 3 (log x/x²) via AntitoneOn.tsum_comp_add_le_integral (N := 3): ∑' n, f(n+3+1) = ∑' n, f(n+4) ✓ with f x = log x/x², anti on Ici 3, integrable on Ioi 3, nonneg on Ioi 3.\n- The integral = (log 3 + 1)/3 via integral_Ioi_of_hasDerivAt_of_tendsto' with F = −(log x + 1)/x → 0.\n- The numeric fold: log 2/4 + log 3/9 + (log 3+1)/3 ≤ 1: with log 2 ≤ 0.6932 (Real.log_two_lt_d9 : log 2 < 0.6931471808), log 3 ≤ 1.1 (via e^1.1 ≥ 3.00126 partial sum): 0.6932/4 + 1.1/9 + (1.1+1)/3 = 0.1733 + 0.1223 + 0.7 = 0.9956 ≤ 1 ✓.\n\nThe antitone condition for f(x) = log x/x² on Ici 3: via deriv ≤ 0 — needs deriv of log x/x² = (1−2log x)/x³ ≤ 0 for x ≥ 3 (log 3 ≥ 1.0986 ≥ 1/2 — needs log 3 ≥ 1/2: from log 3 ≥ log e^{1/2}... e^{1/2} ≤ 1.65 ≤ 3 ✓ via exp(1/2) partial sum: e^{0.5} ≤ ... hmm Real.exp_one_lt_d9 : e < 2.7182818286; e^{1/2} ≤ 1.65: via exp partial sums: e^{0.5} ≤ Σ_{k≤4} 0.5^k/k! + tail: 1+0.5+0.125+0.0208+0.0026+... ≈ 1.6487 ≤ 1.65 ✓ — the Lean route: Real.exp_le_of... there's `Real.exp_one_lt_d9` for e¹; for e^{0.5} there's... `Real.exp_half_...`? Hmm. Alternative: log 3 ≥ 1: 3 ≥ e ⟸ Real.exp_one_lt_d9 (e ≤ 2.7183 ≤ 3) + Real.log_le_log (0 < e) (e ≤ 3): log e = 1 ≤ log 3 ✓✓ — `Real.log_le_iff_le_exp`? or Real.log_exp: log (exp 1) = 1 ≤ log 3 via Real.log_le_log (exp_pos 1) (by linarith [Real.exp_one_lt_d9.le] : exp 1 ≤ 3) ✓ clean!). So log 3 ≥ 1 ⟹ for x ≥ 3: 2 log x ≥ 2 ⟹ 1 − 2 log x ≤ −1 < 0 ⟹ deriv f x = (1 − 2 log x)/x³ ≤ 0 ⟹ antitone via antitoneOn_of_deriv_nonpos (convex (Ici 3)) (continuousOn f) (differentiableOn interior) (deriv ≤ 0). ContinuousOn f: log x/x² continuous on Ici 3 (x ≠ 0) ✓ via ContinuousOn.div (Real.continuousOn_log... (Real.continuousOn_log : ContinuousOn log {0}ᶜ) hmm — `Real.continuousOn_log : ContinuousOn Real.log {0}ᶜ` ✓ I believe). DifferentiableOn: similar. deriv f x: deriv (fun x => log x/x²) x = ... via deriv_div (hasDerivAt...). ~50 lines.\n\nThe Ioi-FTC: integral_Ioi_of_hasDerivAt_of_tendsto' (hderiv : ∀ x ∈ Ici a, HasDerivAt F (f' x) x) (f'int) (hlim : Tendsto F atTop (𝓝 m)) : ∫ Ioi a f' = m − F a. With F = fun x => −(log x + 1)/x, f' x = log x/x²: \n- hderiv: HasDerivAt (fun x => −(log x + 1)/x) (log x/x²) x for x ≥ 3: from HasDerivAt.div of (log + 1) and id:\n  (hasDerivAt_log'?). Real.hasDerivAt_log (hx : x ≠ 0) : HasDerivAt log (1/x) x ✓. (.add_const 1) : HasDerivAt (fun x => log x + 1) (1/x) x ✓ (HasDerivAt.add_const). Then .div (hasDerivAt_id' x) (hx0 : x ≠ 0)... HasDerivAt.div (hc : HasDerivAt c c' x) (hd : HasDerivAt d d' x) (hd0 : d x ≠ 0) : HasDerivAt (fun y => c y / d y) ((c' * d x − c x * d') / (d x)^2) x. With c = log+1, c' = 1/x, d = id, d' = 1: ((1/x)·x − (log x + 1)·1)/x² = (1 − log x − 1)/x² = −log x/x². So the div gives HasDerivAt ((log+1)/id) (−log x/x²) x — then .neg: HasDerivAt (−(log+1)/x) (log x/x²) x ✓. The value form needs convert/field_simp+ring to match log x / x^2. The function form: (fun y => (log y + 1)/y) vs (fun x => −(log x + 1)/x).neg — (fun y => −((log y+1)/y)) vs (fun y => −(log y + 1)/x)... should be fine via congr/ring_nf.\n- f'int: IntegrableOn (fun x => log x/x²) (Ioi 3): via `integrableOn_Ioi_deriv_of_nonneg'` — wait — from the example: integrableOn_Ioi_deriv_of_nonneg' (c) (F) (hlim : Tendsto F atTop (𝓝 0))? (hderiv : ∀ x ∈ Ioi c, HasDerivAt F (f x) x) (hnonneg : ∀ x ∈ Ioi c, 0 ≤ f x): IntegrableOn f (Ioi c) — hmm the example called `integrableOn_Ioi_deriv_of_nonneg' _ _ tendsto_log_atTop.inv_tendsto_atTop.neg` then two goals (hasDerivAt, nonneg). So signature: integrableOn_Ioi_deriv_of_nonneg' (a : ℝ) (F : ℝ → ℝ) (hF : Tendsto F atTop (𝓝 0)) (hderiv : ∀ x ∈ Ioi a, ...) (hpos : ∀ x ∈ Ioi a, 0 ≤ f' x)? Roughly. I'll mimic.\n- hlim: Tendsto (fun x => −(log x + 1)/x) atTop (𝓝 0): (log x + 1)/x = log x/x + 1/x: tendsto: Real.tendsto_log_div_atTop? — check name: I believe `Real.tendsto_log_div_atTop : Tendsto (fun x => Real.log x / x) atTop (𝓝 0)` — hmm might not exist by that name; there IS `Real.tendsto_pow_div_exp...`? For log x / x → 0: it's a special case of `Real.tendsto_log_div_rpow...`? or `tendsto_div_log_...`? There's `Real.tendsto_log_div_atTop` — I'm fairly confident (in Analysis/SpecialFunctions/Log/Deriv... or Limits). If missing: log x/x = (1/x)·log x — bound |log x/x| ≤ 2/√x·... via log x ≤ 2√x hmm — or Real.tendsto_log_div_atTop exists. Check at build. Then (log x + 1)/x = log x/x + 1/x → 0 + 0: Tendsto.add + tendsto_one_div_atTop... `tendsto_one_div_atTop_nhds_zero`? hmm: Tendsto (fun x => 1/x) atTop (𝓝 0) = `tendsto_one_div_atTop_nhds_zero` ✓-ish. Then .neg for the outer minus.\n\nOK — I have the full plan for piece 1. Let me also plan piece 6 (digamma) since it's the other 5f-ii piece, then decide what to build tonight. Actually — time: it's ~1:45 AM. Piece 1 is ~150 lines. Let me build piece 1 now, then checkpoint. Piece 3 and 6 are the big ones for 5f-ii.\n\nActually — wait. Let me reconsider whether to include piece 1 tonight. It's self-contained but involves the integral plumbing. Let me try — if it gets stuck, I'll bank the state and leave it for the next session. Let me write piece 1 into LiTightR.lean (§7).","signature":""},{"type":"text","text":"The summability route is clear (`ArithmeticFunction.LSeriesSummable_vonMangoldt` + `summable_norm_iff`). Writing piece 1 (dirichletC2 ≤ 1) — the integral-test tail with the clean antiderivative −(log x + 1)/x."},{"type":"tool_use","id":"Bash_160","name":"Bash","input":{"command":"M=D:/Python/Quantum/zeta-23-lean/.lake/packages/mathlib/Mathlib; grep -rn \"theorem tendsto_log_div_atTop\\|tendsto_log_div_atTop\" $M --include=\"*.lean\" | head -2; grep -rn \"theorem cpow_natCast\\|cpow_natCast\" $M/Analysis/SpecialFunctions/Pow/Complex.lean | head -2; grep -rn \"theorem.*summable_nat_add_iff\" $M --include=\"*.lean\" | head -2; grep -rn \"integrableOn_Ioi_deriv_of_nonneg'\" $M --include=\"*.lean\" | head -3","description":"Verify tendsto_log_div_atTop, cpow_natCast, summable_nat_add_iff, integrableOn_Ioi_deriv_of_nonneg'"},"extra_content":null}]