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In Wikipedia's proof of Riemann's functional equation for the zeta function (here, and click "Show Proof"), I find the assertion that

$$\sum _{n=-\infty }^{\infty } e^{-n^2 \pi x} = \frac{1}{\sqrt{x}}\sum _{n=-\infty }^{\infty } e^{-\frac{n^2 \pi }{x}}$$

I can't work out how this works. Is it to do with Jacobi theta functions?

Mathematica (which seems to use Jacobi's original notation) simplifies the expression on the left hand side above to the Jacobi elliptic theta function (Wikipedia here, plus the 'Auxiliary Functions' section that follows)

$$ \begin{aligned} \sum _{n=-\infty }^{\infty } e^{-n^2 \pi x} &= \vartheta_{3}(0,e^{-\pi x}) \\&= \vartheta_{00}(0,e^{-\pi x}) \\&= \vartheta(0,e^{-\pi x}) \end{aligned} $$

Wikipedia defines

$$\vartheta(z;\tau) := \sum _{n=-\infty }^{\infty } e^{\pi i n^2 \tau+2 \pi i n z}$$

But setting $z = 0$ and $\tau = e^{-\pi x}$ then gives

$$\vartheta(0,e^{-\pi x}) = \sum _{n=-\infty }^{\infty } e^{\pi i n^2 e^{-\pi x}}$$

which is clearly not equivalent to the original expression.

I suspect that this may have something to do with nomes, which Wikipedia mentions but I cannot get my head around.

So, my two questions are:

  1. How do I prove the original equivalence?
  2. What am I doing wrong in relation to the Jacobi theta function?