Let $f(z)=\sum_{-\infty}^\infty e^{2\pi inz}e^{-\pi n^2}$. Show there is a unique $z_0$ in the unit square such that $f(z_0)=0$.
How to prove $f(z) = \sum_{n=-\infty}^{\infty}e^{2\pi inz}e^{-\pi n^2}$ has a unique zero inside a unit square in the 1st quadrant.