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.

For a cusp form f of level Γ(n), we have t^s * f(it) → 0 as t → ∞ for any real power s. This follows from the exponential decay of cusp forms at infinity: f = O(exp(-2π τ.im / n)).

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

D(exp(cz))/exp(cz) = c/(2πi) for any coefficient c.