← Session 29 — ii′-5e-v: circle bound (far-tail + series bounds)
thinking (3654 chars)
Good. Key signatures:
- `logDeriv_riemannZeta_sub_nearZeros_le {t : ℝ} (ht : 10 ≤ |t|) {s : ℂ} (hsim : s.im = t) (hσlo : 1/2 ≤ s.re) (hσhi : s.re ≤ 7/2) (hs0 : riemannZeta s ≠ 0) : ‖logDeriv ζ s − Σ_{ρ ∈ nearZeros t} (zeroMult ρ:ℂ)/(s−ρ)‖ ≤ 100000·(log|t|)²`
- `logDeriv_riemannZeta_sub_nearZeros_le_left {t} (ht : 10 ≤ |t|) {s} (hsim : s.im = t) (hσlo : −5/2 ≤ s.re) (hσhi : s.re ≤ 1/2) (hs0 : ζ s ≠ 0) : ... ≤ 200000·(log|t|)²`
- `nearZeros t := (zeroDiscFinset t).filter fun ρ => |ρ.im − t| < 1` — a Finset ℂ. zeroDiscFinset t — the zeros in the disc around 2+it. Elements: zeros (IsNontrivialZero? zeroDisc_isNontrivial — from session 023: zeroDisc_isNontrivial).
- `norm_digamma_sub_log_le_of_two_le_abs_im {w : ℂ} (hw : 2 ≤ |w.im|) : ‖Complex.digamma w − Complex.log w‖ ≤ 3`.
- `logDeriv_zeta_fe` — needs 10 hypotheses (hs0, hs1, hG : Gammaℝ s ≠ 0, hζ : ζ s ≠ 0, hd : DifferentiableAt Gammaℝ s, and the same at 1−s).
Now check LiDigammaLeft's (A), (B), (G) and LiReflect's norm-log lemma, plus the nearZeros count input (zeroDiscFinset's count — for the near sum I need Σ_{ρ ∈ nearZeros t} zeroMult ρ: is there a count? Or I bound via the im-band brick on the image... the nearZeros t elements are ℂ (not the subtype) — my brick is for the subtype finset. The nearZeros count: Ncount(t−1, t+1]-ish — hmm: nearZeros t ⊆ zerosIn(t−1, t+1) ∪ zerosIn(−t−1, −t+1)? The elements ρ ∈ nearZeros t have |γ−t| < 1: if γ > 0: γ ∈ (t−1, t+1) — hmm t could be negative! t = s.im ∈ [−R, R]. The brick works on the subtype with the band a ≤ |γ| ≤ b: |γ| ∈ (|t|−1, |t|+1): a := |t|−1, b := |t|+1, c := ... |γ| ≥ |t|−1 ≥ 9 ≥ 1 ✓. But nearZeros t is a Finset ℂ — the subtype finset: map? The mult-sum Σ_{ρ ∈ nearZeros t} (zeroMult ρ : ℝ) — I can bound via the brick applied to a subtype finset whose image ⊆ nearZeros... OR simpler: use Ncount directly: nearZeros t's elements with γ > 0 are in zerosIn(t−1, t+1]; with γ < 0 in zerosIn(−t−1, −t+1]... but nearZeros is a Finset ℂ and the Ncount bridge (finsum_mem_eq_finite_toFinset_sum) works on zerosIn's toFinset: Σ_{ρ ∈ nearZeros t} m ≤ Σ_{W1.toFinset ∪ W2.toFinset} m = Ncount + Ncount — the same image-subset pattern as the brick but with nearZeros t directly (no Subtype.val image needed — it's already ℂ)! Even simpler than the brick. Then Ncount(t−1, t+1] ≤ A₀·3·log(3·max|t−1||t+1|+5) (interval_gen: b−a+1 = 3) and Ncount(−t−1, −t+1] similarly: total ≤ 6·A₀·log(3(|t|+1)+5).
Wait — are all elements of nearZeros t nontrivial zeros? zeroDisc_isNontrivial (session 023: zeroDisc_isNontrivial : ρ ∈ zeroDiscFinset t → IsNontrivialZero ρ? something like that — yes: "zeroDiscFinset + zeroDisc_isNontrivial (re ≥ 1 via Mathlib's one_le_re nonvanishing; re ≤ 0 via ΓR-pole/FE)"). So zerosIn membership OK (zerosIn requires IsNontrivialZero).
Also — do I know |γ − t| < 1 ⟹ sign analysis: γ ∈ (t−1, t+1): if t > 0: γ > t−1 ≥ 9 > 0 ✓ all in W1?? WAIT — if t ≥ 10: γ ∈ (t−1, t+1) ⊆ (9, ...): γ > 0 ✓ — then ALL nearZeros are in zerosIn(t−1, t+1] — only ONE window needed! Similarly t ≤ −10: γ ∈ (t−1, t+1) ⊆ (−11, −9): all negative: zerosIn(t−1, t+1] = zerosIn(t−1, t+1] with both negative — one window! So the count is ONE Ncount(t−1, t+1] ≤ A₀·3·log(3·max|t−1||t+1|+5). Even simpler. (t = s.im fixed sign.) ✓
But — Ncount counts zerosIn (a, b] — membership needs a < γ ≤ b: |γ−t| < 1 ⟹ t−1 < γ < t+1 ⟹ γ ∈ (t−1, t+1) ⊆ (t−1, t+1] ✓ (non-strict upper fine).
So near-count: Σ_{ρ ∈ nearZeros t} (zeroMult ρ : ℝ) ≤ Ncount(t−1, t+1] ≤ 3·A₀·log(3·max|t−1||t+1|+5) — via sum_le_sum_of_subset_of_nonneg + the finsum bridge. Need nearZeros t ⊆ (zerosIn (t−1) (t+1)).toFinset. ✓
Now the (A)/(B)/(G) from LiDigammaLeft:
Now the 5e-ii digamma API and the norm-log lemma from LiReflect.
tool: Grep ok
input
{
"pattern": "(theorem norm_digamma_le_of_two_le_re|theorem norm_digamma_le_of_dist_nat|theorem logDeriv_Gamma\u211d_eq|theorem norm_log_le|lemma norm_log_le)",
"path": "D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1",
"output_mode": "content",
"-n": true,
"-A": 10
}
result 2026-08-24 01:47:30Z → 2026-08-24 01:47:30Z
mode='content' num_files=0 filenames=[] content="research\\lean\\NguyenRH1\\NguyenRH1\\LiReflect.lean:415:theorem norm_log_le_of_one_le {w : ℂ} (hw : 1 ≤ ‖w‖) :\nresearch\\lean\\NguyenRH1\\NguyenRH1\\LiReflect.lean-416- ‖Complex.log w‖ ≤ Real.log ‖w‖ + Real.pi:= by\nresearch\\lean\\NguyenRH1\\NguyenRH1\\LiReflect.lean-417- calc ‖Complex.log w‖ ≤ |(Complex.log w).re| + |(Complex.log w).im|:=\nD:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiReflect.lean-418- norm_le_abs_re_add_abs_im' _\nresearch\\lean\\NguyenRH1\\NguyenRH1\\LiReflect.lean-419- _ = |Real.log ‖w‖| + |Complex.arg w|:= by rw [Complex.log_re, Complex.log_im]\nresearch\\lean\\NguyenRH1\\NguyenRH1\\LiReflect.lean-420- _ = Real.log ‖w‖ + |Complex.arg w|:= by rw [abs_of_nonneg (Real.log_nonneg hw)]\nresearch\\lean\\NguyenRH1\\NguyenRH1\\LiReflect.lean-421- _ ≤ Real.log ‖w‖ + Real.pi:= add_le_add_right (Complex.abs_arg_le_pi w) _\nD:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiReflect.lean-422-\nresearch\\lean\\NguyenRH1\\NguyenRH1\\LiReflect.lean-423-\\-- Stirling-flavoured digamma bound on the vertical strip:\nD:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiReflect.lean-424-`‖ψ(w)‖ ≤ log|t| + 5` when `|im w| = |t|/2`, `re w > 0`, `‖w‖ ≤ 2|t|`. -/\nresearch\\lean\\NguyenRH1\\NguyenRH1\\LiReflect.lean-425-theorem norm_digamma_le_log_of {t: ℝ} (ht : 10 ≤ |t|) {w : ℂ}\n--\nresearch\\lean\\NguyenRH1\\NguyenRH1\\LiDigammaLeft.lean:198:theorem norm_digamma_le_of_two_le_re {w : ℂ} (hwre : 2 ≤ w.re) (hwim : |w.im| ≤ 5) :\nresearch\\lean\\NguyenRH1\\NguyenRH1\\LiDigammaLeft.lean-199- ‖Complex.digamma w‖ ≤ 20 * Real.log (2 + ‖w‖):= by\nresearch\\lean\\NguyenRH1\\NguyenRH1\\LiDigammaLeft.lean-200- have hw_norm_ge: 2 ≤ ‖w‖ := by\nresearch\\lean\\NguyenRH1\\NguyenRH1\\LiDigammaLeft.lean-201- have h1:= Complex.abs_re_le_norm w\nD:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiDigammaLeft.lean-202- rw [abs_of_nonneg (by linarith [hwre])] at h1\nD:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiDigammaLeft.lean-203- linarith [hwre, h1]\nresearch\\lean\\NguyenRH1\\NguyenRH1\\LiDigammaLeft.lean-204- have h1log4: Real.log 4 ≤ Real.log (2 + ‖w‖) :=\nD:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiDigammaLeft.lean-205- Real.log_le_log (by norm_num) (by linarith [hw_norm_ge])\nresearch\\lean\\NguyenRH1\\NguyenRH1\\LiDigammaLeft.lean-206- have hlog4gt: (1 : ℝ) < Real.log 4 := by\nresearch\\lean\\NguyenRH1\\NguyenRH1\\LiDigammaLeft.lean-207- have h4:= Real.log_two_gt_d9.le\nresearch\\lean\\NguyenRH1\\NguyenRH1\\LiDigammaLeft.lean-208- have h5: Real.log 4 = 2 * Real.log 2 := by\n--\nresearch\\lean\\NguyenRH1\\NguyenRH1\\LiDigammaLeft.lean:720:theorem norm_digamma_le_of_dist_nat {c : ℝ} (hc : 0 < c) {w : ℂ}\nresearch\\lean\\NguyenRH1\\NguyenRH1\\LiDigammaLeft.lean-721- (hwre: w.re ≤ 2) (hdist : ∀ m : ℕ, c ≤ ‖w + (m : ℂ)‖) :\nresearch\\lean\\NguyenRH1\\NguyenRH1\\LiDigammaLeft.lean-722- ‖Complex.digamma w‖ ≤ (46 + 2 \\ c) + 23 * Real.log (2 + ‖w‖):= by\nD:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiDigammaLeft.lean-723- -- pole-avoidance from the dist condition\nresearch\\lean\\NguyenRH1\\NguyenRH1\\LiDigammaLeft.lean-724- have hpoles: ∀ k : ℕ, w ≠ -(k : ℂ) := by\nD:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiDigammaLeft.lean-725- intro k h\nresearch\\lean\\NguyenRH1\\NguyenRH1\\LiDigammaLeft.lean-726- have h1:= hdist k\nD:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiDigammaLeft.lean-727- rw [h] at h1\nresearch\\lean\\NguyenRH1\\NguyenRH1\\LiDigammaLeft.lean-728- rw [show (-(k: ℂ)) + (k : ℂ) = 0 by ring, norm_zero] at h1\nD:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiDigammaLeft.lean-729- linarith [hc, h1]\nresearch\\lean\\NguyenRH1\\NguyenRH1\\LiDigammaLeft.lean-730- -- the shift count: lands re(w + m) ∈ [3, 4)\n--\nresearch\\lean\\NguyenRH1\\NguyenRH1\\LiDigammaLeft.lean:911:theorem logDeriv_Gammaℝ_eq {s : ℂ} (hs : s.im ≠ 0) :\nD:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiDigammaLeft.lean-912- logDeriv Complex.Gammaℝ s\nresearch\\lean\\NguyenRH1\\NguyenRH1\\LiDigammaLeft.lean-913- = -(Real.log Real.pi: ℂ) / 2 + (1 / 2 : ℂ) * Complex.digamma (s / 2) := by\nresearch\\lean\\NguyenRH1\\NguyenRH1\\LiDigammaLeft.lean-914- set m:= ⌈max 0 (-(s.re / 2))⌉₊ + 1 with hm_def\nresearch\\lean\\NguyenRH1\\NguyenRH1\\LiDigammaLeft.lean-915- have hmge: (2 : ℝ) ≤ s.re + 2 * (m : ℝ) := by\nresearch\\lean\\NguyenRH1\\NguyenRH1\\LiDigammaLeft.lean-916- have h1: max 0 (-(s.re / 2)) ≤ ((⌈max 0 (-(s.re / 2))⌉₊ : ℕ) : ℝ) := Nat.le_ceil _\nresearch\\lean\\NguyenRH1\\NguyenRH1\\LiDigammaLeft.lean-917- have h2: -(s.re / 2) ≤ max 0 (-(s.re / 2)) := le_max_right _ _\nD:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiDigammaLeft.lean-918- rw [hm_def]\nD:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiDigammaLeft.lean-919- push_cast\nD:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiDigammaLeft.lean-920- linarith [h1, h2]\nresearch\\lean\\NguyenRH1\\NguyenRH1\\LiDigammaLeft.lean-921- have hres2: (0 : ℝ) < (s + 2 * (m : ℂ)).re := by" num_lines=47 num_matches=None applied_limit=None applied_offset=None
raw content_json (database row)
[{"type":"thinking","thinking":"Good. Key signatures:\n- `logDeriv_riemannZeta_sub_nearZeros_le {t : ℝ} (ht : 10 ≤ |t|) {s : ℂ} (hsim : s.im = t) (hσlo : 1/2 ≤ s.re) (hσhi : s.re ≤ 7/2) (hs0 : riemannZeta s ≠ 0) : ‖logDeriv ζ s − Σ_{ρ ∈ nearZeros t} (zeroMult ρ:ℂ)/(s−ρ)‖ ≤ 100000·(log|t|)²`\n- `logDeriv_riemannZeta_sub_nearZeros_le_left {t} (ht : 10 ≤ |t|) {s} (hsim : s.im = t) (hσlo : −5/2 ≤ s.re) (hσhi : s.re ≤ 1/2) (hs0 : ζ s ≠ 0) : ... ≤ 200000·(log|t|)²`\n- `nearZeros t := (zeroDiscFinset t).filter fun ρ => |ρ.im − t| < 1` — a Finset ℂ. zeroDiscFinset t — the zeros in the disc around 2+it. Elements: zeros (IsNontrivialZero? zeroDisc_isNontrivial — from session 023: zeroDisc_isNontrivial).\n- `norm_digamma_sub_log_le_of_two_le_abs_im {w : ℂ} (hw : 2 ≤ |w.im|) : ‖Complex.digamma w − Complex.log w‖ ≤ 3`.\n- `logDeriv_zeta_fe` — needs 10 hypotheses (hs0, hs1, hG : Gammaℝ s ≠ 0, hζ : ζ s ≠ 0, hd : DifferentiableAt Gammaℝ s, and the same at 1−s).\n\nNow check LiDigammaLeft's (A), (B), (G) and LiReflect's norm-log lemma, plus the nearZeros count input (zeroDiscFinset's count — for the near sum I need Σ_{ρ ∈ nearZeros t} zeroMult ρ: is there a count? Or I bound via the im-band brick on the image... the nearZeros t elements are ℂ (not the subtype) — my brick is for the subtype finset. The nearZeros count: Ncount(t−1, t+1]-ish — hmm: nearZeros t ⊆ zerosIn(t−1, t+1) ∪ zerosIn(−t−1, −t+1)? The elements ρ ∈ nearZeros t have |γ−t| < 1: if γ > 0: γ ∈ (t−1, t+1) — hmm t could be negative! t = s.im ∈ [−R, R]. The brick works on the subtype with the band a ≤ |γ| ≤ b: |γ| ∈ (|t|−1, |t|+1): a := |t|−1, b := |t|+1, c := ... |γ| ≥ |t|−1 ≥ 9 ≥ 1 ✓. But nearZeros t is a Finset ℂ — the subtype finset: map? The mult-sum Σ_{ρ ∈ nearZeros t} (zeroMult ρ : ℝ) — I can bound via the brick applied to a subtype finset whose image ⊆ nearZeros... OR simpler: use Ncount directly: nearZeros t's elements with γ > 0 are in zerosIn(t−1, t+1]; with γ < 0 in zerosIn(−t−1, −t+1]... but nearZeros is a Finset ℂ and the Ncount bridge (finsum_mem_eq_finite_toFinset_sum) works on zerosIn's toFinset: Σ_{ρ ∈ nearZeros t} m ≤ Σ_{W1.toFinset ∪ W2.toFinset} m = Ncount + Ncount — the same image-subset pattern as the brick but with nearZeros t directly (no Subtype.val image needed — it's already ℂ)! Even simpler than the brick. Then Ncount(t−1, t+1] ≤ A₀·3·log(3·max|t−1||t+1|+5) (interval_gen: b−a+1 = 3) and Ncount(−t−1, −t+1] similarly: total ≤ 6·A₀·log(3(|t|+1)+5).\n\nWait — are all elements of nearZeros t nontrivial zeros? zeroDisc_isNontrivial (session 023: zeroDisc_isNontrivial : ρ ∈ zeroDiscFinset t → IsNontrivialZero ρ? something like that — yes: \"zeroDiscFinset + zeroDisc_isNontrivial (re ≥ 1 via Mathlib's one_le_re nonvanishing; re ≤ 0 via ΓR-pole/FE)\"). So zerosIn membership OK (zerosIn requires IsNontrivialZero).\n\nAlso — do I know |γ − t| < 1 ⟹ sign analysis: γ ∈ (t−1, t+1): if t > 0: γ > t−1 ≥ 9 > 0 ✓ all in W1?? WAIT — if t ≥ 10: γ ∈ (t−1, t+1) ⊆ (9, ...): γ > 0 ✓ — then ALL nearZeros are in zerosIn(t−1, t+1] — only ONE window needed! Similarly t ≤ −10: γ ∈ (t−1, t+1) ⊆ (−11, −9): all negative: zerosIn(t−1, t+1] = zerosIn(t−1, t+1] with both negative — one window! So the count is ONE Ncount(t−1, t+1] ≤ A₀·3·log(3·max|t−1||t+1|+5). Even simpler. (t = s.im fixed sign.) ✓\n\nBut — Ncount counts zerosIn (a, b] — membership needs a < γ ≤ b: |γ−t| < 1 ⟹ t−1 < γ < t+1 ⟹ γ ∈ (t−1, t+1) ⊆ (t−1, t+1] ✓ (non-strict upper fine).\n\nSo near-count: Σ_{ρ ∈ nearZeros t} (zeroMult ρ : ℝ) ≤ Ncount(t−1, t+1] ≤ 3·A₀·log(3·max|t−1||t+1|+5) — via sum_le_sum_of_subset_of_nonneg + the finsum bridge. Need nearZeros t ⊆ (zerosIn (t−1) (t+1)).toFinset. ✓\n\nNow the (A)/(B)/(G) from LiDigammaLeft:","signature":""},{"type":"text","text":"Now the 5e-ii digamma API and the norm-log lemma from LiReflect."},{"type":"tool_use","id":"Grep_143","name":"Grep","input":{"pattern":"(theorem norm_digamma_le_of_two_le_re|theorem norm_digamma_le_of_dist_nat|theorem logDeriv_Gammaℝ_eq|theorem norm_log_le|lemma norm_log_le)","path":"D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1","output_mode":"content","-n":true,"-A":10},"extra_content":null}]