Define the unit square $0 \leq \Re(z) \leq 1$, $0 \leq \Im(z) \leq 1$. Using the argument principle, show there is a unique $z_0$ in the unit square such that $f(z_0)=0$ for $f(z)=\sum_{-\infty}^\infty e^{2\pi inz}e^{-\pi n^2}$.
Just by looking at the series, I have deduced that $f(z+1) =f(z)$ and $f(z+i) = e^\pi e^{2 \pi i z} f(z)$ for all $z$.
Since $f(z)$ doesn't vanish on the unit square, we may apply the Argument Principle. Between the two vertical lines of the square, $f$ stays on the real line. On the horizontal lines, the endpoints are the same.
How do we deduce the change in the argument over the horizontal lines? Obviously, the series in question has some relation to fourier analysis, but I don't know anything about that. What other properties of this series can we use to make use of the Argument Principle?
