Plain-language statement
E₂ q-expansion in sigma form: E₂ = 1 - 24 * ∑ σ₁(n) * q^n. This follows from G2_q_exp and the definition E₂ = (1/(2*ζ(2))) • G₂. The proof expands the definitions and simplifies using ζ(2) = π²/6.
Exact Lean statement
lemma E₂_sigma_qexp (z : UpperHalfPlane) :
E₂ z = 1 - 24 * ∑' (n : ℕ+), (ArithmeticFunction.sigma 1 n : ℂ) *
Complex.exp (2 * Real.pi * Complex.I * n * z)Formal artifact
Lean source
lemma E₂_sigma_qexp (z : UpperHalfPlane) : E₂ z = 1 - 24 * ∑' (n : ℕ+), (ArithmeticFunction.sigma 1 n : ℂ) * Complex.exp (2 * Real.pi * Complex.I * n * z) := by -- Use E₂_eq and tsum_eq_tsum_sigma to convert n*q^n/(1-q^n) → σ₁(n)*q^n rw [E₂_eq z] congr 2 -- Convert between ℕ+ and ℕ indexing using tsum_pnat_eq_tsum_succ have hl := tsum_pnat_eq_tsum_succ (f := fun n => ArithmeticFunction.sigma 1 n * Complex.exp (2 * π * Complex.I * n * z)) have hr := tsum_pnat_eq_tsum_succ (f := fun n => n * Complex.exp (2 * π * Complex.I * n * z) / (1 - Complex.exp (2 * π * Complex.I * n * z))) rw [hl, hr] have ht := tsum_eq_tsum_sigma z simp at * rw [ht]- Project
- Sphere Packing in Dimension 8
- License
- Apache-2.0
- Commit
- acfc6204e65a
- Source
- SpherePacking/ModularForms/FG.lean:220-235
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Related declarations
Anti Der Pos
antiDerPos
Plain-language statement
If is a modular form where is positive for sufficiently large (i.e. constant term is positive) and the derivative is positive, then is also positive.
Source project: Sphere Packing in Dimension 8
Person-level attribution pending.
Anti Serre Der Pos
antiSerreDerPos
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
Plain-language statement
Closed ball centered at z with radius z.im/2 is contained in the upper half plane.
Source project: Sphere Packing in Dimension 8
Person-level attribution pending.