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I'm trying to understand the proof of the functional equation for the L-series of primitive, even Dirichlet characters. For even, primitive characters we have $$\theta_\chi(x):=\sum_{n\in \mathbb{Z}} \chi(n)\exp\left(\frac{-\pi}{q}n^2x^2\right).$$

In my lecture notes it says

The theta function decays exponentially as $x \rightarrow \infty$ [$\theta_\chi(x)=O(e^{-\pi/qx^2})$]

which I assume means $\theta_\chi(x)=O\left(\exp\left(\frac{-\pi}{q}x^2\right)\right)$ since I want to deduce that the integral

$$\int_1^\infty \theta_\chi(x)x^s\frac{dx}{x} $$

converges.

I "feel" like this is vaguely because when $n$ gets big, the $\exp\left(\frac{-\pi}{q}n^2x^2\right)$ is so small that is contributes basically nothing to the sum. But I don't really think this is enough, since

$$\int_{-\infty}^\infty \exp\left(\frac{-\pi}{q}x^2\xi^2\right)d\xi \leq \sum_{n \in \mathbb{Z}}\exp\left(\frac{-\pi}{q}n^2x^2\right) \leq 2+ \int_{-\infty}^\infty \exp\left(\frac{-\pi}{q}x^2\xi^2\right)d\xi$$

so

$$\frac{\sqrt{q}}{|x|} \leq \sum_{n \in \mathbb{Z}}\exp\left(\frac{-\pi}{q}n^2x^2\right) \leq 2+\frac{\sqrt{q}}{|x|}$$

which is not strong enough. So how do I prove that

The theta function decays exponentially as $x \rightarrow \infty$ [$\theta_\chi(x)=O(e^{-\pi/qx^2})$]

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Single out $n = \pm 1$ term, and start to use integral comparison starting from $n = 2$. –  Sanchez Apr 25 '13 at 8:52
    
@Sanchez I singled out the $\pm1$ term like you suggested, but couldn't figure out how to finish it off using integral comparison (I don't know how to evaluate that integral except doing it over all of $\mathbb{R}$). Is my answer below okay? –  porkramen Apr 25 '13 at 10:32
    
Yes :) ........ –  Sanchez Apr 25 '13 at 16:56
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1 Answer

So, using Sanchez's hint, I think I have

$$\begin{array}{rll}\theta_\chi(x) & \leq & 2\exp\left(\frac{-\pi}{q}x^2\right)+\sum_{|n|\geq 2}\exp\left( \frac{-\pi}{q}n^2x^2\right) \\ & \leq & 2\exp\left(\frac{-\pi}{q}x^2\right)+\sum_{|n|\geq 4}\exp\left( \frac{-\pi}{q}n x^2\right) \\ \text{(expanding the sum} \\ \text{ as a geometric series)} & \leq &2 \exp\left(\frac{-\pi}{q}x^2\right)+ 2\exp \left(\frac{-\pi}{q}4x^2\right)\left( 1-\exp\left(\frac{-\pi}{q} x^2\right)\right)^{-1}\end{array}$$

which does it.

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