Given $\Theta(\tau)=\sum_{n \in \mathbb Z}exp(2\pi in² \tau)$ and $\tau \in \mathbb H$
I am trying to prove the following identity:
$\Theta(-\frac{1}{2\pi})=\sqrt{\frac{\tau}{i}}\Theta(\frac{\tau}{2})$
Expanding the sum I get the following:
$\Theta(-\frac{1}{2\pi})=\sum_{n \in \mathbb Z}exp(-in² \tau)$
Might be a dumb question but how do I get the factor $\sqrt{\frac{\tau}{i}}$ from that sum
Would appreciate any help
Best Answer
As currently written, your question is malformed. We can see this because $$ \Theta(-1/2\pi)$$ is a constant, while $$ \sqrt{\frac{\tau}{i}} \Theta(\tau/2)$$ is a (nonconstant) function of $\tau$.
It is likely that what you meant was to prove that $$\Theta(i/\tau) = \sqrt{\tau} \Theta(i\tau),$$ or an equivalent transformation. This is a classically studied transformation, used by Riemann in one of his proofs the functional equation of $\zeta(s)$. This is traditionally proved through Poisson summation on the definition of $\Theta(\tau)$.