[{"type":"thinking","thinking":"Committed. Now continue with piece 1 (dirichletC2 ≤ 0.995) — the integral bound. This is the remaining elementary piece; pieces 3, 6 go to 5f-ii.\n\nPiece 1 plan:\n```lean\ntheorem dirichletC2_le : dirichletC2 ≤ 0.995\n```\ndirichletC2 := ∑' n, ‖LSeries.term (Λ) (2:ℂ) n‖.\n\nStep A: rewrite the term norm: for n ≥ 1, ‖LSeries.term Λ 2 n‖ = Λ(n)/n² (real); for n = 0: LSeries.term f s 0 = 0 (if n = 0 → 0). So dirichletC2 = ∑' n : ℕ, (if n = 0 then 0 else (Λ(n))/n²).\n\nLSeries.term def: `LSeries.term (f : ℕ → ℂ) (s : ℂ) (n : ℕ) : ℂ = f n / n ^ s`? with 0 for n=0? Let me check: LSeries.term f s n := if n = 0 then 0 else f n / n ^ s — I believe there's a `LSeries.term_of_ne_zero`/`LSeries.term_of_pos`. Let me check the def.\n\nStep B: bound Λ(n) ≤ log n: `ArithmeticFunction.vonMangoldt_le_log : Λ n ≤ log n`? Check name — I recall `ArithmeticFunction.vonMangoldt_le_log (n : ℕ) : Λ n ≤ Real.log n`? Hmm — might be `vonMangoldt_le_log` in Mathlib/NumberTheory/VonMangoldt.lean. My earlier grep of VonMangoldt.lean for \"vonMangoldt_le_log\\|vonMangoldt_le\" returned nothing (the file may be named differently). Let me search.\n\nStep C: the tsum bound: ∑' n, (if n = 0 then 0 else log n / n²) ≤ 0.995. Route: tsum_eq_sum over {0,1,2,3} head + tail via AntitoneOn.tsum_comp_add_le_integral... The head terms: n=0: 0; n=1: log 1/1 = 0; n=2: log 2/4; n=3: log 3/9. Tail: Σ_{n≥4} log n/n² ≤ ∫₃^∞ log x/x² = (log 3+1)/3.\n\nHmm — the AntitoneOn.tsum_le_integral family: `AntitoneOn.tsum_le_integral (anti : AntitoneOn f (Ici 0)) (integrable : IntegrableOn f (Ioi 0)) (nonneg) : ∑' n, f n ≤ f 0 + ∫ Ioi 0` — the integrability requirement (Ioi 0) is global — for f = log x/x² on Ici 3 only. The `tsum_comp_add_le_integral` variant: ∑' n, f (n + N) ≤ ∫ Ioi N — with anti/integrable on Ioi N... Let me read the exact signature of tsum_comp_add_le_integral.\n\nActually — maybe simpler: the comparison via a CRUDE per-term bound that's still summable and computable: for n ≥ 4, log n ≤ n^{1/2}·(2/e)?? — computed before as lossy. The integral route is needed. Let me read the tsum_comp_add_le_integral signature.","signature":""},{"type":"text","text":"Piece 1 (dirichletC2 ≤ 0.995) next. Checking the integral-comparison signature and the vonMangoldt bound."},{"type":"tool_use","id":"Bash_155","name":"Bash","input":{"command":"M=D:/Python/Quantum/zeta-23-lean/.lake/packages/mathlib/Mathlib; sed -n '215,235p' $M/Analysis/SumIntegralComparisons.lean; echo ===VM===; grep -rn \"vonMangoldt_le_log\\|vonMangoldt_nonneg\" $M/NumberTheory/*.lean | head -5; echo ===TERM===; grep -rn \"def term\\|theorem term\" $M/NumberTheory/LSeries/Basic.lean | head -8","description":"Check tsum_comp_add_le_integral, vonMangoldt bounds, LSeries.term"},"extra_content":null}]