Research proof index

The vanishing order of F at infinity is 2. Blueprint: F = 720² * q² * (1 + O(q)), so F / q² → 720² as im(z) → ∞.

F₁ has exponential decay at infinity (it's essentially D E₄ which decays).

$\lim_{t \to 0^+} F(it) / G(it) = 18 / \pi^2$. 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.