← Session 33 — ii′-3b/3c crux + F-D finding; Lagarias letter package
thinking (1825 chars)
I left a sorry in the zetaA(−2) = 0 part (the (−2) ∈ Sz forward direction) and the proof has some shaky spots (the `(−2 : ℂ).re ∈ uIcc` via simp, the `h0` absurd case, the hm1' proof). Let me fix the sorry first: ζ(−2) = 0 — trivial zeros: Mathlib `riemannZeta_neg_two`? Hmm — does Mathlib have ζ(−2) = 0? There's `riemannZeta_neg_nat`? Let me think: the trivial zeros at negative even integers: Mathlib has `riemannZeta_neg_two`? Not sure. Better: zetaA(−2) = 0 via zetaA_apply: (−2−1)·ζ(−2): need ζ(−2) = 0.
Alternative: use completedRiemannZeta: Λ(−2) = ΓR(−2)·ζ(−2) — ΓR has a pole at −2... hmm. Direct: Mathlib has `riemannZeta_neg_two`? Let me check — actually in LiTrivialZeros (5e-0), zeroMult_neg_two_mul proves zeroMult(−2n) = 1 for all n ≥ 1 — which requires ζ(−2n) = 0 with order 1! So there's machinery: LiTrivialZeros' zeroMult_neg_two (ii′-4) and zeroMult_neg_two_mul (5e-0) — zeroMult(−2) = 1 implies ζ(−2) = 0 (analyticOrderAt = 1 ≠ 0,∞). Use that: from zeroMult_neg_two : zeroMult (-2) = 1, get riemannZeta (-2) = 0: zeroMult is analyticOrderNatAt? zeroMult ρ := (analyticOrderAt riemannZeta ρ).toNat (check def in Zeta23). From zeroMult(-2) = 1 and the order being 1: analyticOrderAt ζ (−2) = 1 ⟹ ζ(−2) = 0 (analyticOrderAt_eq_natCast... the order is 1 only if ζ(−2) = 0 with order 1). Simpler: look at how zeroMult_neg_two's proof establishes ζ(−2) = 0 — it used the factorization ζ = (s+2)·Λ/Hfun with Hfun(−2) ≠ 0, Λ(−2) = Λ(3) ≠ 0 — that gives ζ(−2) = 0 directly! But is there a standalone lemma `riemannZeta_neg_two_eq_zero`? Let me grep LiTrivialZeros/LiContour for a direct ζ(−2n) = 0.
Actually — in LiContour's zeroMult_neg_two proof, the factorization ζ = (s+2)·Λ/Hfun near −2 directly gives ζ(−2) = 0·(Λ/Hfun)(−2) = 0. Is there a lemma? Let me grep for `riemannZeta (-2` and `riemannZeta_neg`.
Fixing the sorry I left: ζ(−2) = 0 should follow from existing machinery. Checking.
tool: Grep ok
input
{
"pattern": "riemannZeta_neg_two|riemannZeta \\(-2|zeta_neg_two|zeroMult_neg_two",
"path": "D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1",
"output_mode": "content",
"-n": true,
"head_limit": 20
}
result 2026-08-25 05:50:45Z → 2026-08-25 05:50:45Z
mode='content' num_files=0 filenames=[] content="NguyenRH1\\LiAssembly.lean:226: have hζ2 : riemannZeta (-2 : ℂ) = 0 := by\nNguyenRH1\\LiAssembly.lean:228: have h2zeros : riemannZeta (-2 : ℂ) = 0 := by\nNguyenRH1\\LiAssembly.lean:229: rw [riemannZeta_neg_two]\nNguyenRH1\\LiContour.lean:701:theorem zeroMult_neg_two : zeroMult (-2) = 1 := by\nNguyenRH1\\LiContour.lean:752: have hz0 : riemannZeta (-2) = 0 := by\nNguyenRH1\\LiContour.lean:753: have h := riemannZeta_neg_two_mul_nat_add_one 0\nNguyenRH1\\LiContour.lean:797: have hzeta_an : AnalyticAt ℂ riemannZeta (-2) := hRHS_an.congr (EventuallyEq.symm hEv)\nNguyenRH1\\LiContour.lean:798: have horder : analyticOrderAt riemannZeta (-2) = 1 :=\nNguyenRH1\\LiTrivialZeros.lean:4:generalizing ii′-4c's `zeroMult_neg_two` (LiContour.lean).\nNguyenRH1\\LiTrivialZeros.lean:13:* §3 `zeroMult_neg_two_mul`: ζ(s) = (s+2n)·(Λ(s)/HfunT n s) near −2n\nNguyenRH1\\LiTrivialZeros.lean:210:ii′-4c's `zeroMult_neg_two`): zeroMult(−2n) = 1 for all n ≥ 1. The local\nNguyenRH1\\LiTrivialZeros.lean:214:theorem zeroMult_neg_two_mul (n : ℕ) (hn : 1 ≤ n) :\nNguyenRH1\\LiTrivialZeros.lean:281: have hz0 : riemannZeta (-2 * (n : ℂ)) = 0 := by\nNguyenRH1\\LiTrivialZeros.lean:282: have h := riemannZeta_neg_two_mul_nat_add_one (n - 1)\nNguyenRH1\\LiTrivialZeros.lean:336: have hzeta_an : AnalyticAt ℂ riemannZeta (-2 * (n : ℂ)) :=\nNguyenRH1\\LiTrivialZeros.lean:338: have horder : analyticOrderAt riemannZeta (-2 * (n : ℂ)) = 1 :=\nNguyenRH1\\LiFEntire.lean:36:* s = −2n: logDeriv ζ − 1/(s+2n) analytic (NF + `zeroMult_neg_two_mul`,\nNguyenRH1\\LiFEntire.lean:472:/-- ζ(−2n) = 0 for n ≥ 1 (from `zeroMult_neg_two_mul`: the order is 1). -/\nNguyenRH1\\LiFEntire.lean:473:lemma riemannZeta_neg_two_mul {n : ℕ} (hn : 1 ≤ n) :\nNguyenRH1\\LiFEntire.lean:474: riemannZeta (-2 * (n : ℂ)) = 0 := by" num_lines=20 num_matches=None applied_limit=20 applied_offset=None
raw content_json (database row)
[{"type":"thinking","thinking":"I left a sorry in the zetaA(−2) = 0 part (the (−2) ∈ Sz forward direction) and the proof has some shaky spots (the `(−2 : ℂ).re ∈ uIcc` via simp, the `h0` absurd case, the hm1' proof). Let me fix the sorry first: ζ(−2) = 0 — trivial zeros: Mathlib `riemannZeta_neg_two`? Hmm — does Mathlib have ζ(−2) = 0? There's `riemannZeta_neg_nat`? Let me think: the trivial zeros at negative even integers: Mathlib has `riemannZeta_neg_two`? Not sure. Better: zetaA(−2) = 0 via zetaA_apply: (−2−1)·ζ(−2): need ζ(−2) = 0. \n\nAlternative: use completedRiemannZeta: Λ(−2) = ΓR(−2)·ζ(−2) — ΓR has a pole at −2... hmm. Direct: Mathlib has `riemannZeta_neg_two`? Let me check — actually in LiTrivialZeros (5e-0), zeroMult_neg_two_mul proves zeroMult(−2n) = 1 for all n ≥ 1 — which requires ζ(−2n) = 0 with order 1! So there's machinery: LiTrivialZeros' zeroMult_neg_two (ii′-4) and zeroMult_neg_two_mul (5e-0) — zeroMult(−2) = 1 implies ζ(−2) = 0 (analyticOrderAt = 1 ≠ 0,∞). Use that: from zeroMult_neg_two : zeroMult (-2) = 1, get riemannZeta (-2) = 0: zeroMult is analyticOrderNatAt? zeroMult ρ := (analyticOrderAt riemannZeta ρ).toNat (check def in Zeta23). From zeroMult(-2) = 1 and the order being 1: analyticOrderAt ζ (−2) = 1 ⟹ ζ(−2) = 0 (analyticOrderAt_eq_natCast... the order is 1 only if ζ(−2) = 0 with order 1). Simpler: look at how zeroMult_neg_two's proof establishes ζ(−2) = 0 — it used the factorization ζ = (s+2)·Λ/Hfun with Hfun(−2) ≠ 0, Λ(−2) = Λ(3) ≠ 0 — that gives ζ(−2) = 0 directly! But is there a standalone lemma `riemannZeta_neg_two_eq_zero`? Let me grep LiTrivialZeros/LiContour for a direct ζ(−2n) = 0.\n\nActually — in LiContour's zeroMult_neg_two proof, the factorization ζ = (s+2)·Λ/Hfun near −2 directly gives ζ(−2) = 0·(Λ/Hfun)(−2) = 0. Is there a lemma? Let me grep for `riemannZeta (-2` and `riemannZeta_neg`.","signature":""},{"type":"text","text":"Fixing the sorry I left: ζ(−2) = 0 should follow from existing machinery. Checking."},{"type":"tool_use","id":"Grep_264","name":"Grep","input":{"pattern":"riemannZeta_neg_two|riemannZeta \\(-2|zeta_neg_two|zeroMult_neg_two","path":"D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1","output_mode":"content","-n":true,"head_limit":20},"extra_content":null}]