How do I show that for a fixed $\omega$ in the upper half plane, the theta function
$\theta(z) = \sum_{n=-\infty}^{\infty} e^{\pi i(n^2\omega + 2nz)}$
is not identically zero? Is there an obvious choice of z that would work or would I have to resort to some fourier coefficient argument?
Thanks
Letting $\omega=iy$ and $y\to +\infty$ gives limit $1$ (from the $n=0$ term), for arbitrary $\omega$. So it's not identically $0$. To prove it's not constant, subtract the $n=0$ term, divide by $e^{2\pi i n\omega}$, and let $\omega=iy$ with $y\to+\infty$ again, getting the $n=1$ coefficient.
But, yes, the Fourier series argument is instantaneous, so is better in the long run, even if one does want to see the guts of things in the short run.
(Also, you might consider interchanging the roles of $\omega$ and $z$, or letting $\omega$ by $\tau$, but this is inessential.)
EDIT: Sorry, due to my expectations about notational choices, I read it as though the roles of $\omega$ and $z$ were reversed from the question. To fix the questioner's $\omega$ and ask whether the resulting function of $z$ can be identically $0$, surely a Fourier series argument is optimal: a Fourier series with not-all-$0$ Fourier coefficients is not identically $0$ as a function (assuming the coefficients decay a bit, so that it has a good point-wise sense).
For generic $\omega$ in the upper half-plane, I don't think there's any clever choice of $z$ to see pointwise-non-zero-ness directly.