← Session 24 — ii′-2…4: swaps, (4.10) design, contour + website roadmap fixes
thinking (3966 chars)
- `LSeries_zeta_eq_riemannZeta (hs : 1 < s.re) : L ↗ζ s = riemannZeta s` ✓
- `riemannZeta_two : riemannZeta 2 = (π:ℂ)²/6` ✓ (HurwitzZetaValues.lean:226) — so ζ(2) = π²/6 ≈ 1.6449 < 2: |ζ(2)| = π²/6: ‖·‖ of a positive real cast: = π²/6.
Center lower bound plan:
```lean
theorem norm_riemannZeta_two_add_it_ge {t : ℝ} :
(Real.pi ^ 2 / 6)⁻¹ ≤ ‖riemannZeta (2 + t * Complex.I)‖ := by
-- 1/ζ(s) = L ↗μ s for re s > 1: from LSeries_zeta_mul_Lseries_moebius + LSeries_zeta_eq_riemannZeta
have h1 : riemannZeta (2 + t * Complex.I) * L ↗μ (2 + t * Complex.I) = 1 := by
rw [← LSeries_zeta_eq_riemannZeta (by simp)]
exact LSeries_zeta_mul_Lseries_moebius (by simp) -- re = 2 > 1 ✓
-- ‖L ↗μ s‖ ≤ ∑' ‖μ n / n^s‖ ≤ ∑' n^{-2} = ζ(2) = π²/6
...
```
The bound ‖L ↗μ s‖ ≤ π²/6: L ↗μ s = ∑' n, (↗μ n : ℂ)/n^s (LSeries def): ‖∑'‖ ≤ ∑' ‖·‖ (Summable version: `Summable.norm_tsum_le_tsum_norm`? or `tsum_norm_le`? there's `Summable.norm_tsum_le`? hmm: ‖∑' f‖ ≤ ∑' ‖f‖ when ∑' ‖f‖ summable: `Summable.tsum_norm_le`? the name might be `Summable.norm_tsum_le_tsum...` — there's `tsum_le_tsum` for reals and for norms... `Summable.norm_tsum_le`? Let me use: ‖∑' n, f n‖ ≤ ∑' n, ‖f n‖ via `Summable.norm_tsum_le_tsum_norm`? Hmm. I'll check the name at build; alternatives: `tsum_norm_le`? Whatever.
Then ‖(↗μ n : ℂ)/n^s‖ = |μ n|/n² ≤ 1/n² (|μ n| ≤ 1 ✓ `ArithmeticFunction.moebius` abs ≤ 1: |μ n| ∈ {0,1} — `ArithmeticFunction.abs_moebius_le_one`? exists probably). And ∑' 1/n² = π²/6: hasSum_zeta_two (HasSum (fun n => 1/n²) (π²/6)) — wait that's for n : ℕ including n = 0 (1/0² = 0 junk ✓ fine) — hasSum_zeta_two : HasSum (fun n : ℕ => 1/(n:ℝ)^2) (π²/6) ✓. So ∑' 1/n² = π²/6 ✓.
So ‖1/ζ(s)‖ ≤ π²/6 ⟹ ‖ζ(s)‖ ≥ 1/(π²/6) = 6/π² ≈ 0.608 ✓ (via ‖1/ζ‖ = 1/‖ζ‖ and the inverse flip: 1/‖ζ‖ ≤ π²/6 ⟹ ‖ζ‖ ≥ 6/π² — via one_div_le... the norm of 1/ζ: ‖1/ζ(s)‖ = 1/‖ζ(s)‖ (norm_div norm_one) — then 1/‖ζ‖ ≤ π²/6 ⟹ ‖ζ‖ ≥ 6/π²: `one_div_le...`?: from 1/‖ζ‖ ≤ C: ‖ζ‖ ≥ 1/C: `one_div_le_one_div_of_le`? I have 1/‖ζ‖ ≤ π²/6 and want 6/π² ≤ ‖ζ‖: rewrite: 1/‖ζ‖ ≤ π²/6 ⟺ 6/π² ≤ ‖ζ‖ (both pos): `div_le_iff`... For the Jensen application, the fz_bound needs ‖F w‖ ≤ B with F = ζ(s₀ + (5/2)w)/ζ(s₀): ‖F w‖ = ‖ζ‖/‖ζ(s₀)‖ ≤ (3/4)|t|²/(6/π²) = (3/4)(π²/6)|t|² = π²|t|²/8. So B := (π²/8)·|t|².
ZerosBound then: Σ_{zeros of F in ball(0, r)} mult ≤ log B/log(R/r): with r = 23/25 (zeros with |w| ≤ 23/25 ⟺ |ρ−s₀| ≤ (5/2)(23/25) = 23/10 = 2.3 ✓), R = 24/25: log(R/r) = log(24/23) ≈ 0.0426: M ≤ (π²|t|²/8)/0.0426·... hmm: log B = log(π²/8) + 2log|t| ≤ 0.21 + 2 log|t|: M ≤ (2 log|t| + 0.21)/0.0426 ≈ 47 log|t| + 5. OK.
Hmm wait — actually the ZerosBound is stated with (hf0_eq_one : f 0 = 1) — F(0) = ζ(s₀)/ζ(s₀) = 1 ✓ (ζ(s₀) ≠ 0 ✓).
Also ZerosBound's SetOfZeros 1 F — zeros of F in the UNIT closedBall — finite: F's zeros = {w : s₀+(5/2)w ∈ zeros of ζ} ∩ ball — finite via the ζ-side finiteness + the affine map... The set {w | ‖w‖ ≤ 1 ∧ F w = 0} = preimage under w ↦ s₀+(5/2)w of {zeros in closedBall(s₀, 5/2)}: finite ⟸ the ζ-side finite (zeroDiscFinset t) via the injective affine map: `Set.Finite.preimage`? with injectivity of w ↦ s₀+(5/2)w ✓ (5/2 ≠ 0). OK.
The scaling relation: analyticOrderNatAt F w = analyticOrderNatAt ζ (s₀ + (5/2)w): order under affine precomposition — same (the order is invariant under the affine change with nonzero slope — `analyticOrderAt_comp`? hmm — provable via the normal form: F(w) = ζ(s₀+cw) = (cw)^m·h(s₀+cw) = c^m·w^m·h(...): order m ✓. There might be a lemma; else prove via the choose-normal-form. Moderate.)
And then the COLLECTION zeros vs the F-zeros: the collection is on the ζ-side (ρ ∈ B(s₀, r_c)) — the mult-weighted count Σ mₖ = Σ over ζ-zeros of analyticOrderNatAt ζ ρ = Σ over F-zeros of analyticOrderNatAt F w (via the order invariance) = ZerosBound's LHS ✓.
OK — this all works but it's a chain of moderate-length steps. Given the session length, let me keep going and write §2 (center lower bound + Jensen count + R-gap). Write it now.
All confirmed (`riemannZeta_two = π²/6` is in Mathlib). Writing §2: center lower bound + Jensen count + the R-gap.
tool: Edit ok
input
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"replace_all": false,
"file_path": "D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiDisc.lean",
"old_string": "end NguyenRH1\n\nend",
"new_string": "/-! ## 2. The center lower bound, the Jensen count, and the R-gap -/\n\n/-- The M\u00f6bius L-series is bounded by \u03b6(2) on re s \u2265 2: |1/\u03b6(s)| \u2264 \u03c0\u00b2/6. -/\ntheorem norm_inv_riemannZeta_le {s : \u2102} (hs : 2 \u2264 s.re) :\n \u2016(riemannZeta s)\u207b\u00b9\u2016 \u2264 Real.pi ^ 2 / 6 := by\n have hs1 : 1 < s.re := by linarith\n have hprod : riemannZeta s * L \u2197ArithmeticFunction.moebius s = 1 := by\n rw [\u2190 LSeries_zeta_eq_riemannZeta hs1]\n exact LSeries_zeta_mul_Lseries_moebius hs1\n have hsum : Summable fun n : \u2115 => \u2016((ArithmeticFunction.moebius n : \u2124) : \u2102) / (n : \u2102) ^ s\u2016 := by\n have h2 : Summable fun n : \u2115 => (1 : \u211d) / (n : \u211d) ^ 2 := hasSum_zeta_two.summable\n apply h2.of_norm_bounded (fun n => ?_)\n rw [norm_div]\n have h3 : \u2016((ArithmeticFunction.moebius n : \u2124) : \u2102)\u2016 \u2264 1 := by\n rw [\u2190 Int.cast_abs]\n apply Int.cast_le.mpr\n exact ArithmeticFunction.abs_moebius_le_one... sorry\n sorry\n sorry\n\nend NguyenRH1\n\nend"
}
result 2026-08-22 19:52:24Z → 2026-08-22 19:52:24Z
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"oldString": "end NguyenRH1\n\nend",
"newString": "/-! ## 2. The center lower bound, the Jensen count, and the R-gap -/\n\n/-- The M\u00f6bius L-series is bounded by \u03b6(2) on re s \u2265 2: |1/\u03b6(s)| \u2264 \u03c0\u00b2/6. -/\ntheorem norm_inv_riemannZeta_le {s : \u2102} (hs : 2 \u2264 s.re) :\n \u2016(riemannZeta s)\u207b\u00b9\u2016 \u2264 Real.pi ^ 2 / 6 := by\n have hs1 : 1 < s.re := by linarith\n have hprod : riemannZeta s * L \u2197ArithmeticFunction.moebius s = 1 := by\n rw [\u2190 LSeries_zeta_eq_riemannZeta hs1]\n exact LSeries_zeta_mul_Lseries_moebius hs1\n have hsum : Summable fun n : \u2115 => \u2016((ArithmeticFunction.moebius n : \u2124) : \u2102) / (n : \u2102) ^ s\u2016 := by\n have h2 : Summable fun n : \u2115 => (1 : \u211d) / (n : \u211d) ^ 2 := hasSum_zeta_two.summable\n apply h2.of_norm_bounded (fun n => ?_)\n rw [norm_div]\n have h3 : \u2016((ArithmeticFunction.moebius n : \u2124) : \u2102)\u2016 \u2264 1 := by\n rw [\u2190 Int.cast_abs]\n apply Int.cast_le.mpr\n exact ArithmeticFunction.abs_moebius_le_one... sorry\n sorry\n sorry\n\nend NguyenRH1\n\nend",
"originalFile": "/-\nNguyenRH1/LiDisc.lean \u2014 tier ii\u2032 sub-stage ii\u2032-5c (D-067 \u00a73 + \u00a76 revision):\nthe LOOSE Davenport-disc R-bound on \u03c3 \u2208 [1/2, 7/2], |t| \u2265 10 \u2014 the\nGROWTH INPUT for the partial-fraction identity (ii\u2032-5e). Its constants\nare deliberately loose (D-067 \u00a76: no disc-only constant can reach the\nbaked A = 10.5, B\u2080 = 48; tightness comes from the identity stage).\n\nStructure (all on the disc B(2+it, R), R \u2208 [2.4, 2.5] chosen in a gap of\nthe zero-distance values):\n* \u00a71 the zero collection + count (perimeter ZeroCountProp, ii\u2032-4);\n* \u00a72 the R-gap selection (scaled midpoint pigeonhole from ii\u2032-4);\n* \u00a73 g := \u03b6/\u220f(s\u2212\u03c1\u2096)^{m\u2096} analytic + nonvanishing on the closed disc;\n* \u00a74 the bounds: |g| on the outer circle (ii\u2032-5b's |\u03b6|-bound +\n clearance), the center lower bound (1/\u03b6 = \u03a3 \u03bc(n)n^{\u2212s}), then\n LogOfAnalyticFunction + BorelCaratheodoryDeriv (StrongPNTPrefix)\n giving \u2016(log g)\u2032\u2016 \u226a (log|t|)\u00b2, hence the loose R-bound.\n\nArchive references: D-067, D-064 \u00a74 R3, journal-022.\n-/\nimport Mathlib\nimport Zeta23.FromPNTPlus.StrongPNTPrefix\nimport NguyenRH1.LiRBound\n\nopen Zeta23 Complex Filter Topology MeasureTheory\nopen scoped Nat\n\nnoncomputable section\n\nnamespace NguyenRH1\n\n/-! ## 1. The zero collection and its count -/\n\n/-- The disc center s\u2080 = 2 + it. -/\ndef discCenter (t : \u211d) : \u2102 := 2 + t * Complex.I\n\n/-- The zeros of \u03b6 in the closed disc of radius 5/2 about 2+it, as a\nfinset (finiteness from Mathlib's discrete-zero-set machinery). -/\nnoncomputable def zeroDiscFinset (t : \u211d) : Finset \u2102 :=\n ((isCompact_closedBall (discCenter t) (5 / 2)).inter_riemannZetaZeros_finite).toFinset\n\ntheorem mem_zeroDiscFinset {t : \u211d} {\u03c1 : \u2102} :\n \u03c1 \u2208 zeroDiscFinset t \u2194\n \u03c1 \u2208 Metric.closedBall (discCenter t) (5 / 2) \u2227 riemannZeta \u03c1 = 0 := by\n unfold zeroDiscFinset\n rw [Set.Finite.mem_toFinset]\n constructor\n \u00b7 intro h\n exact \u27e8h.1, h.2\u27e9\n \u00b7 intro h\n exact \u27e8h.1, mem_riemannZetaZeros.mpr h.2\u27e9\n\n/-- Every zero of \u03b6 in the disc is a nontrivial zero (re \u2265 1 excluded by\n`riemannZeta_ne_zero_of_one_le_re`; re \u2264 0 excluded by the FE since the\ndisc avoids {\u22122n} and 0). -/\ntheorem zeroDisc_isNontrivial {t : \u211d} {\u03c1 : \u2102} (h\u03c1 : \u03c1 \u2208 zeroDiscFinset t) :\n IsNontrivialZero \u03c1 := by\n have h2 := (mem_zeroDiscFinset.mp h\u03c1).2\n have hball := (mem_zeroDiscFinset.mp h\u03c1).1\n -- \u03c1 \u2208 closedBall (2+it) (5/2) \u27f9 re \u03c1 \u2208 [\u22121/2, 9/2]\n have hre : -1 / 2 \u2264 \u03c1.re \u2227 \u03c1.re \u2264 9 / 2 := by\n rw [Metric.mem_closedBall, Complex.dist_eq] at hball\n have h1 : |(\u03c1 - discCenter t).re| \u2264 5 / 2 := (abs_re_le_norm _).trans hball\n have h2re : (\u03c1 - discCenter t).re = \u03c1.re - 2 := by\n simp [discCenter, sub_re, mul_re, ofReal_re, ofReal_im, I_re, I_im]\n rw [h2re] at h1\n have := abs_le.mp h1\n constructor <;> linarith [this.1, this.2]\n -- re \u03c1 \u2265 1 excluded\n have hnot1 : \u03c1.re < 1 := by\n by_contra h\n push Not at h\n exact riemannZeta_ne_zero_of_one_le_re h h2\n -- re \u03c1 \u2264 0 excluded (would force \u03c1 \u2208 {\u22122n} \u222a {0}, all outside the disc)\n have hpos : 0 < \u03c1.re := by\n by_contra h\n push Not at h\n by_cases hG : Complex.Gamma\u211d \u03c1 = 0\n \u00b7 -- \u0393R(\u03c1) = 0 \u27f9 \u03c1 = \u22122n: n = 0 gives \u03b6(0) = \u22121/2 \u2260 0; n \u2265 1 gives re \u2264 \u22122\n rw [Complex.Gamma\u211d_eq_zero_iff] at hG\n obtain \u27e8n, hn\u27e9 := hG\n have hre' : \u03c1.re = -2 * (n : \u211d) := by\n rw [hn]\n simp [neg_re, mul_re]\n rcases Nat.eq_zero_or_pos n with hn0 | hnpos\n \u00b7 subst hn0\n norm_num at hre'\n rw [hn] at h2\n norm_num at h2\n rw [riemannZeta_zero] at h2\n norm_num at h2\n \u00b7 have hn1 : (1:\u211d) \u2264 (n : \u211d) := by exact_mod_cast hnpos\n linarith [hre.1, hre']\n \u00b7 have h\u03c10 : \u03c1 \u2260 0 := by\n intro h0\n rw [h0, riemannZeta_zero] at h2\n norm_num at h2\n have h\u039b : completedRiemannZeta \u03c1 = 0 := by\n rw [RvM.completedRiemannZeta_eq_Gamma\u211d_mul h\u03c10 hG, h2, mul_zero]\n exact absurd (RvM.completedRiemannZeta_eq_zero_iff.mp h\u039b).2.1 (not_lt.mpr h)\n exact \u27e8h2, hpos, hnot1\u27e9\n\nend NguyenRH1\n\nend\n",
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" exact absurd (RvM.completedRiemannZeta_eq_zero_iff.mp h\u039b).2.1 (not_lt.mpr h)",
" exact \u27e8h2, hpos, hnot1\u27e9",
" ",
"+/-! ## 2. The center lower bound, the Jensen count, and the R-gap -/",
"+",
"+/-- The M\u00f6bius L-series is bounded by \u03b6(2) on re s \u2265 2: |1/\u03b6(s)| \u2264 \u03c0\u00b2/6. -/",
"+theorem norm_inv_riemannZeta_le {s : \u2102} (hs : 2 \u2264 s.re) :",
"+ \u2016(riemannZeta s)\u207b\u00b9\u2016 \u2264 Real.pi ^ 2 / 6 := by",
"+ have hs1 : 1 < s.re := by linarith",
"+ have hprod : riemannZeta s * L \u2197ArithmeticFunction.moebius s = 1 := by",
"+ rw [\u2190 LSeries_zeta_eq_riemannZeta hs1]",
"+ exact LSeries_zeta_mul_Lseries_moebius hs1",
"+ have hsum : Summable fun n : \u2115 => \u2016((ArithmeticFunction.moebius n : \u2124) : \u2102) / (n : \u2102) ^ s\u2016 := by",
"+ have h2 : Summable fun n : \u2115 => (1 : \u211d) / (n : \u211d) ^ 2 := hasSum_zeta_two.summable",
"+ apply h2.of_norm_bounded (fun n => ?_)",
"+ rw [norm_div]",
"+ have h3 : \u2016((ArithmeticFunction.moebius n : \u2124) : \u2102)\u2016 \u2264 1 := by",
"+ rw [\u2190 Int.cast_abs]",
"+ apply Int.cast_le.mpr",
"+ exact ArithmeticFunction.abs_moebius_le_one... sorry",
"+ sorry",
"+ sorry",
"+",
" end NguyenRH1",
" ",
" end"
]
}
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raw content_json (database row)
[{"type":"thinking","thinking":"- `LSeries_zeta_eq_riemannZeta (hs : 1 < s.re) : L ↗ζ s = riemannZeta s` ✓\n- `riemannZeta_two : riemannZeta 2 = (π:ℂ)²/6` ✓ (HurwitzZetaValues.lean:226) — so ζ(2) = π²/6 ≈ 1.6449 < 2: |ζ(2)| = π²/6: ‖·‖ of a positive real cast: = π²/6.\n\nCenter lower bound plan:\n```lean\ntheorem norm_riemannZeta_two_add_it_ge {t : ℝ} :\n (Real.pi ^ 2 / 6)⁻¹ ≤ ‖riemannZeta (2 + t * Complex.I)‖ := by\n -- 1/ζ(s) = L ↗μ s for re s > 1: from LSeries_zeta_mul_Lseries_moebius + LSeries_zeta_eq_riemannZeta\n have h1 : riemannZeta (2 + t * Complex.I) * L ↗μ (2 + t * Complex.I) = 1 := by\n rw [← LSeries_zeta_eq_riemannZeta (by simp)]\n exact LSeries_zeta_mul_Lseries_moebius (by simp) -- re = 2 > 1 ✓\n -- ‖L ↗μ s‖ ≤ ∑' ‖μ n / n^s‖ ≤ ∑' n^{-2} = ζ(2) = π²/6\n ...\n```\nThe bound ‖L ↗μ s‖ ≤ π²/6: L ↗μ s = ∑' n, (↗μ n : ℂ)/n^s (LSeries def): ‖∑'‖ ≤ ∑' ‖·‖ (Summable version: `Summable.norm_tsum_le_tsum_norm`? or `tsum_norm_le`? there's `Summable.norm_tsum_le`? hmm: ‖∑' f‖ ≤ ∑' ‖f‖ when ∑' ‖f‖ summable: `Summable.tsum_norm_le`? the name might be `Summable.norm_tsum_le_tsum...` — there's `tsum_le_tsum` for reals and for norms... `Summable.norm_tsum_le`? Let me use: ‖∑' n, f n‖ ≤ ∑' n, ‖f n‖ via `Summable.norm_tsum_le_tsum_norm`? Hmm. I'll check the name at build; alternatives: `tsum_norm_le`? Whatever.\n\nThen ‖(↗μ n : ℂ)/n^s‖ = |μ n|/n² ≤ 1/n² (|μ n| ≤ 1 ✓ `ArithmeticFunction.moebius` abs ≤ 1: |μ n| ∈ {0,1} — `ArithmeticFunction.abs_moebius_le_one`? exists probably). And ∑' 1/n² = π²/6: hasSum_zeta_two (HasSum (fun n => 1/n²) (π²/6)) — wait that's for n : ℕ including n = 0 (1/0² = 0 junk ✓ fine) — hasSum_zeta_two : HasSum (fun n : ℕ => 1/(n:ℝ)^2) (π²/6) ✓. So ∑' 1/n² = π²/6 ✓.\n\nSo ‖1/ζ(s)‖ ≤ π²/6 ⟹ ‖ζ(s)‖ ≥ 1/(π²/6) = 6/π² ≈ 0.608 ✓ (via ‖1/ζ‖ = 1/‖ζ‖ and the inverse flip: 1/‖ζ‖ ≤ π²/6 ⟹ ‖ζ‖ ≥ 6/π² — via one_div_le... the norm of 1/ζ: ‖1/ζ(s)‖ = 1/‖ζ(s)‖ (norm_div norm_one) — then 1/‖ζ‖ ≤ π²/6 ⟹ ‖ζ‖ ≥ 6/π²: `one_div_le...`?: from 1/‖ζ‖ ≤ C: ‖ζ‖ ≥ 1/C: `one_div_le_one_div_of_le`? I have 1/‖ζ‖ ≤ π²/6 and want 6/π² ≤ ‖ζ‖: rewrite: 1/‖ζ‖ ≤ π²/6 ⟺ 6/π² ≤ ‖ζ‖ (both pos): `div_le_iff`... For the Jensen application, the fz_bound needs ‖F w‖ ≤ B with F = ζ(s₀ + (5/2)w)/ζ(s₀): ‖F w‖ = ‖ζ‖/‖ζ(s₀)‖ ≤ (3/4)|t|²/(6/π²) = (3/4)(π²/6)|t|² = π²|t|²/8. So B := (π²/8)·|t|².\n\nZerosBound then: Σ_{zeros of F in ball(0, r)} mult ≤ log B/log(R/r): with r = 23/25 (zeros with |w| ≤ 23/25 ⟺ |ρ−s₀| ≤ (5/2)(23/25) = 23/10 = 2.3 ✓), R = 24/25: log(R/r) = log(24/23) ≈ 0.0426: M ≤ (π²|t|²/8)/0.0426·... hmm: log B = log(π²/8) + 2log|t| ≤ 0.21 + 2 log|t|: M ≤ (2 log|t| + 0.21)/0.0426 ≈ 47 log|t| + 5. OK.\n\nHmm wait — actually the ZerosBound is stated with (hf0_eq_one : f 0 = 1) — F(0) = ζ(s₀)/ζ(s₀) = 1 ✓ (ζ(s₀) ≠ 0 ✓).\n\nAlso ZerosBound's SetOfZeros 1 F — zeros of F in the UNIT closedBall — finite: F's zeros = {w : s₀+(5/2)w ∈ zeros of ζ} ∩ ball — finite via the ζ-side finiteness + the affine map... The set {w | ‖w‖ ≤ 1 ∧ F w = 0} = preimage under w ↦ s₀+(5/2)w of {zeros in closedBall(s₀, 5/2)}: finite ⟸ the ζ-side finite (zeroDiscFinset t) via the injective affine map: `Set.Finite.preimage`? with injectivity of w ↦ s₀+(5/2)w ✓ (5/2 ≠ 0). OK.\n\nThe scaling relation: analyticOrderNatAt F w = analyticOrderNatAt ζ (s₀ + (5/2)w): order under affine precomposition — same (the order is invariant under the affine change with nonzero slope — `analyticOrderAt_comp`? hmm — provable via the normal form: F(w) = ζ(s₀+cw) = (cw)^m·h(s₀+cw) = c^m·w^m·h(...): order m ✓. There might be a lemma; else prove via the choose-normal-form. Moderate.)\n\nAnd then the COLLECTION zeros vs the F-zeros: the collection is on the ζ-side (ρ ∈ B(s₀, r_c)) — the mult-weighted count Σ mₖ = Σ over ζ-zeros of analyticOrderNatAt ζ ρ = Σ over F-zeros of analyticOrderNatAt F w (via the order invariance) = ZerosBound's LHS ✓.\n\nOK — this all works but it's a chain of moderate-length steps. Given the session length, let me keep going and write §2 (center lower bound + Jensen count + R-gap). Write it now.","signature":""},{"type":"text","text":"All confirmed (`riemannZeta_two = π²/6` is in Mathlib). Writing §2: center lower bound + Jensen count + the R-gap."},{"type":"tool_use","id":"Edit_347","name":"Edit","input":{"replace_all":false,"file_path":"D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiDisc.lean","old_string":"end NguyenRH1\n\nend","new_string":"/-! ## 2. The center lower bound, the Jensen count, and the R-gap -/\n\n/-- The Möbius L-series is bounded by ζ(2) on re s ≥ 2: |1/ζ(s)| ≤ π²/6. -/\ntheorem norm_inv_riemannZeta_le {s : ℂ} (hs : 2 ≤ s.re) :\n ‖(riemannZeta s)⁻¹‖ ≤ Real.pi ^ 2 / 6 := by\n have hs1 : 1 < s.re := by linarith\n have hprod : riemannZeta s * L ↗ArithmeticFunction.moebius s = 1 := by\n rw [← LSeries_zeta_eq_riemannZeta hs1]\n exact LSeries_zeta_mul_Lseries_moebius hs1\n have hsum : Summable fun n : ℕ => ‖((ArithmeticFunction.moebius n : ℤ) : ℂ) / (n : ℂ) ^ s‖ := by\n have h2 : Summable fun n : ℕ => (1 : ℝ) / (n : ℝ) ^ 2 := hasSum_zeta_two.summable\n apply h2.of_norm_bounded (fun n => ?_)\n rw [norm_div]\n have h3 : ‖((ArithmeticFunction.moebius n : ℤ) : ℂ)‖ ≤ 1 := by\n rw [← Int.cast_abs]\n apply Int.cast_le.mpr\n exact ArithmeticFunction.abs_moebius_le_one... sorry\n sorry\n sorry\n\nend NguyenRH1\n\nend"},"extra_content":null}]