As the title reads, a homework problem in fourier analysis is to show the above equality.
The function $N(n) = \{(n_1, n_2) \in \mathbb{Z}^2 : n_1^2 + n_2^2 = n\}$ is the number of points in $\mathbb{Z}^2 $ that have distance exactly $\sqrt{n}$ to the origin.
We are following the book fourier series and integrals by Dym & Mckean.
The book introduces the "theta function": For $t>0 , \theta(t) = \sum_{n = -\infty}^{ +\infty} e^{-\pi n^2 t}$
I have tried to analyse this using the similar swindle used to prove the Jacobi theta function identity:(p52 in the book) $\theta(t) = \frac{1}{\sqrt{t}}\theta(\frac{1}{t})$, however, I can not make any progress. Nor do I see where the properties of the function $N(n)$ are to be used. I am aware that there is a closed form expression of $N(n)$, nevetheless I was unable to use that in the setting of a fourier series.
Moreover, I am assuming that $t>0$ but this information is not given. Naturally the proposition holds for $t=1$. Any hints would be appreciated!