Research proof index

f₄ tends to 0 at infinity. Proof: f₄ = serre_D 2 H₄ + (1/6)H₄(2H₂ + H₄) serre_D 2 H₄ = D H₄ - (1/6)E₂ H₄ → 0 - (1/6)11 = -1/6 (since H₄ → 1, E₂ → 1) H₄(2H₂ + H₄) → 1*(0 + 1) = 1 So f₄ → -1/6 + (1/6)*1 = 0.

$\lim_{t \to 0^+} F(it) / G(it) = 18 / \pi^2$. Proof outline (following blueprint Lemma 8.8): 1. Change of variables: lim_{t→0⁺} F(it)/G(it) = lim_{s→∞} F(i/s)/G(i/s) 2. Apply functional equations: - F(i/s) = s^12F(is) - 12s^11/πF₁(is)E₄(is) + 36s^10/π²E₄(is)² - G(i/s) = s^10H₄(is)³(2H₄(is)² + 5H₄(is)*H₂(is) + 5H₂(is)²) 3. Divide to get: F(i/s)/G(i/...

FmodGReal is differentiable on (0, ∞).

G(1/s) = s^10 * (H₄(is))³ * (2(H₄(is))² + 5H₂(is)H₄(is) + 5(H₂(is))²)

Functional equation of $G$ restricted to the imaginary axis.

G / q^(3/2) → 20480 as im(z) → ∞. Here q^(3/2) = exp(2πi · (3/2) · z).