Research proof index

If $F$ is a modular form where $F(it)$ is positive for sufficiently large $t$ (i.e. constant term is positive) and the derivative is positive, then $F$ is also positive.

Let $F : \mathbb{H} \to \mathbb{C}$ be a holomorphic function where $F(it)$ is real for all $t > 0$. Assume that Serre derivative $\partial_k F$ is positive on the imaginary axis. If $F(it)$ is positive for sufficiently large $t$, then $F(it)$ is positive for all $t > 0$.

Closed ball centered at z with radius z.im/2 is contained in the upper half plane.

Basic properties of derivatives: linearity, Leibniz rule, etc.

The D-derivative is bounded at infinity for bounded holomorphic functions. For y large (y ≥ 2·max(A,0) + 1), we use a ball of radius z.im/2 around z. The ball lies in the upper half plane, f is bounded by M on it, and norm_D_le_of_sphere_bound gives ‖D f z‖ ≤ M/(π·z.im) ≤ M/π.

Lemma 6.45 (Blueprint): The normalized derivative $D$ acts as $q \frac{d}{dq}$ on $q$-series. For a single q-power term: D(a·qⁿ) = n·a·qⁿ where q = exp(2πiz) and n ∈ ℤ. The key calculation: - d/dz(exp(2πinz)) = 2πin·exp(2πinz) - D(exp(2πinz)) = (2πi)⁻¹·(2πin·exp(2πinz)) = n·exp(2πinz)