TITLE: How to evaluate $\int_0^x\vartheta_3(0,t)\ dt$?
QUESTION [6 upvotes]: Due to this question, I found myself trying to take the following integral:
$$\int_0^x\vartheta_3(0,t)\ dt=\ ?$$
However, I know not of how to do this.  As per this post, I find the evaluation at $x=1$ to be $\frac\pi{\tanh\pi}$.  It is equivalent to trying to evaluate the following series:
$$\sum_{n=0}^\infty\frac{x^{n^2}}{n^2+1}$$
My end problem is that I want to evaluate the following integral:
$$\int_0^1\frac1x\int_0^x\vartheta_3(0,t)\ dt\ dx=\sum_{n=0}^\infty\frac1{(n^2+1)^2}$$

REPLY [5 votes]: About the last point, we have:
$$ \mathcal{L}^{-1}\left(\frac{1}{(x^2+1)^2}\right)=\frac{\sin s-s\cos s}{2} $$
hence:
$$\sum_{n\geq 0}\frac{1}{(n^2+1)^2} = 1+\int_{0}^{+\infty}\frac{\sin s-s\cos s}{2(e^s-1)}\,ds$$
and the last integral can be computed through the residue theorem. As an alternative, starting from
$$ \frac{\sinh(\pi x)}{\pi x}=\prod_{n\geq 1}\left(1+\frac{x^2}{n^2}\right)$$
and considering $\frac{d}{dx}\log(\cdot)$ of both sides, we get:
$$ -\frac{1}{x}+\pi\coth(\pi x) = \sum_{n\geq 1}\frac{2x}{n^2+x^2} $$
and by differentiating again:
$$ \frac{1}{x^2}-\frac{\pi^2}{\sinh^2(\pi x)}=2\sum_{n\geq 1}\frac{n^2-x^2}{(n^2+x^2)^2}$$
so that:
$$\boxed{ \sum_{n\geq 1}\frac{1}{(n^2+x^2)^2} = \color{red}{\frac{1}{4x^4}\left(-2+\pi x\coth(\pi x)+\left(\frac{\pi x}{\sinh(\pi x)}\right)^2\right)}}$$
We may also exploit the Poisson summation formula to get the same, since the Fourier transform of $\frac{1}{(1+x^2)^2}$ is a multiple of $(1+|x|)\,e^{-|x|}$.