Theta series of integer squares
= Theta series of integer squares
{title2=$\theta(\tau)=\sum_{n\in\mathbb Z}e^{2\pi in^2\tau}$}
This convention for the <Jacobi theta function> is the usual theta constant evaluated at twice the argument. The <Poisson summation formula> gives $\theta(-1/(4\tau))=\sqrt{-2i\tau}\,\theta(\tau)$, with the square root holomorphic on the half-plane and positive on the imaginary axis after evaluating its positive real argument.