← Session 24 — ii′-2…4: swaps, (4.10) design, contour + website roadmap fixes
thinking (637 chars)
Now §2. Let me structure it:
```lean
/-! ## 2. The center lower bound, the Jensen count, and the R-gap -/
```
First the center lower bound (needed for Jensen):
1/ζ(s) = Σ μ(n)n^{−s} for re s > 1. Lemma names to check: LSeries_zeta_mul_Lseries_moebius : L ↗ζ s * L ↗μ s = 1 (re > 1). And L ↗ζ s = riemannZeta s for re > 1 — the name? `ArithmeticFunction.LSeries_zeta_eq_riemannZeta`? Let me check quickly, plus the ζ(2) ≤ 2 bound availability.
Then the bound: ‖1/ζ(s)‖ = ‖L ↗μ s‖ ≤ ∑' ‖μ n / n^s‖ ≤ ∑' 1/n^2 (for re s = 2)... and ∑' 1/n^2 = riemannZeta 2 (via L ↗ζ 2). So ‖ζ(s)‖ ≥ 1/ζ(2) ≥ 1/2 (if ζ(2) ≤ 2).
Let me check those names.
Now §2. Checking the L-series names for the center lower bound.
tool: Bash ok
input
{
"command": "cd D:/Python/Quantum/zeta-23-lean/.lake/packages/mathlib && grep -rn \"LSeries_zeta_eq\\|zeta_eq_riemannZeta\\|LSeries.*riemannZeta\" Mathlib/NumberTheory/LSeries/Dirichlet.lean Mathlib/NumberTheory/LSeries/Basic.lean 2>/dev/null | grep -i \"theorem\\|lemma\" | head -6; grep -rn \"riemannZeta_two\\|zeta_two\\|hasSum.*1 /.*\\^ 2\" Mathlib/NumberTheory/ -r --include=*.lean | grep -i \"theorem\\|lemma\" | head -5",
"description": "Find LSeries-zeta and zeta(2) evaluation lemmas"
}
result 2026-08-22 19:51:02Z → 2026-08-26 08:05:53Z
Try this:
[apply] ring_nf
The `ring` tactic failed to close the goal. Use `ring_nf` to obtain a normal form.
Note that `ring` works primarily in *commutative* rings. If you have a noncommutative ring, abelian group or module, consider using `noncomm_ring`, `abel` or `module` instead.
NguyenRH1/LiDogLeg.lean:739:59: error: unsolved goals
n : ℕ
t : ℝ
hn : 16 ≤ n
ht : ↑n ≤ t ^ 2
hnR : 16 ≤ ↑n
hsq : 0 < √↑n
⊢ 2 * ↑√↑n + ↑t * I - 1 = 2 * ↑√↑n - 1 + ↑t * I
NguyenRH1/LiDogLeg.lean:756:4: error: No goals to be solved
NguyenRH1/LiDogLeg.lean:777:24: error(lean.unknownIdentifier): Unknown identifier `one_div_mul`
Try this:
[apply] ring_nf
The `ring` tactic failed to close the goal. Use `ring_nf` to obtain a normal form.
Note that `ring` works primarily in *commutative* rings. If you have a noncommutative ring, abelian group or module, consider using `noncomm_ring`, `abel` or `module` instead.
NguyenRH1/LiDogLeg.lean:805:69: error: unsolved goals
n : ℕ
t : ℝ
hn : 16 ≤ n
ht : ↑n ≤ t ^ 2
hnR : 16 ≤ ↑n
hsq : 0 < √↑n
hw1 : ↑(2 * √↑n) + ↑t * I - 1 = ↑(2 * √↑n - 1) + ↑t * I
raw content_json (database row)
[{"type":"thinking","thinking":"Now §2. Let me structure it:\n\n```lean\n/-! ## 2. The center lower bound, the Jensen count, and the R-gap -/\n```\n\nFirst the center lower bound (needed for Jensen):\n1/ζ(s) = Σ μ(n)n^{−s} for re s > 1. Lemma names to check: LSeries_zeta_mul_Lseries_moebius : L ↗ζ s * L ↗μ s = 1 (re > 1). And L ↗ζ s = riemannZeta s for re > 1 — the name? `ArithmeticFunction.LSeries_zeta_eq_riemannZeta`? Let me check quickly, plus the ζ(2) ≤ 2 bound availability.\n\nThen the bound: ‖1/ζ(s)‖ = ‖L ↗μ s‖ ≤ ∑' ‖μ n / n^s‖ ≤ ∑' 1/n^2 (for re s = 2)... and ∑' 1/n^2 = riemannZeta 2 (via L ↗ζ 2). So ‖ζ(s)‖ ≥ 1/ζ(2) ≥ 1/2 (if ζ(2) ≤ 2).\n\nLet me check those names.","signature":""},{"type":"text","text":"Now §2. Checking the L-series names for the center lower bound."},{"type":"tool_use","id":"Bash_346","name":"Bash","input":{"command":"cd D:/Python/Quantum/zeta-23-lean/.lake/packages/mathlib && grep -rn \"LSeries_zeta_eq\\|zeta_eq_riemannZeta\\|LSeries.*riemannZeta\" Mathlib/NumberTheory/LSeries/Dirichlet.lean Mathlib/NumberTheory/LSeries/Basic.lean 2>/dev/null | grep -i \"theorem\\|lemma\" | head -6; grep -rn \"riemannZeta_two\\|zeta_two\\|hasSum.*1 /.*\\^ 2\" Mathlib/NumberTheory/ -r --include=*.lean | grep -i \"theorem\\|lemma\" | head -5","description":"Find LSeries-zeta and zeta(2) evaluation lemmas"},"extra_content":null}]