Plain-language statement
The q-expansion identity E₂E₄ - E₆ = 720·Σn·σ₃(n)·qⁿ. This follows from Ramanujan's formula: E₂E₄ - E₆ = 3·D(E₄), combined with D(E₄) = 240·Σn·σ₃(n)·qⁿ (since D multiplies q-coefficients by n).
Exact Lean statement
theorem E₂_mul_E₄_sub_E₆ (z : ℍ) :
(E₂ z) * (E₄ z) - (E₆ z) = 720 * ∑' (n : ℕ+), n * (σ 3 n) * cexp (2 * π * Complex.I * n * z)Formal artifact
Lean source
theorem E₂_mul_E₄_sub_E₆ (z : ℍ) : (E₂ z) * (E₄ z) - (E₆ z) = 720 * ∑' (n : ℕ+), n * (σ 3 n) * cexp (2 * π * Complex.I * n * z) := by -- From ramanujan_E₄: D E₄ = (1/3) * (E₂ * E₄ - E₆) -- So: E₂ * E₄ - E₆ = 3 * D E₄ have hRam : (E₂ z) * (E₄ z) - (E₆ z) = 3 * D E₄.toFun z := by have h := congrFun ramanujan_E₄ z simp only [Pi.mul_apply, Pi.sub_apply, show (3⁻¹ : ℍ → ℂ) z = 3⁻¹ from rfl] at h field_simp at h ⊢ ring_nf at h ⊢ exact h.symm -- Substitute D(E₄) = 240 * ∑' n, n * σ₃(n) * q^n rw [hRam, DE₄_qexp] ring- Project
- Sphere Packing in Dimension 8
- License
- Apache-2.0
- Commit
- acfc6204e65a
- Source
- SpherePacking/ModularForms/FG.lean:402-415
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Anti Der Pos
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Plain-language statement
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Source project: Sphere Packing in Dimension 8
Person-level attribution pending.
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Plain-language statement
Let be a holomorphic function where is real for all . Assume that Serre derivative is positive on the imaginary axis. If is positive for sufficiently large , then is positive for all .
Source project: Sphere Packing in Dimension 8
Person-level attribution pending.
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closedBall_center_subset_upperHalfPlane
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Source project: Sphere Packing in Dimension 8
Person-level attribution pending.