A generating function for the squares

Jacobi's approach makes use of certain power series which are closely related to the problem of writing a non-negative integer as a sum of squares. One such series is particularly important in this connection:

$$\vartheta_3(0,q) = 1 + \sum_{n=1}^\infty 2 q^{n^2} = 1 + 2q + 2q^4 +2q^9 + 2q^{16} + \dots$$

Consider any non-negative integer $N$. If $N$ is a positive square, then it has exactly two representations, e.g. $N=n^2$ and $N=(-n)^2$ as a square. If $N=0=0^2$, then it has exactly one such representation as a square. And, of course if $N$ is not a square, it has no such representation.

How does this relate to the series $$\vartheta_3(0,q)$$? For any non-negative integer $N$, consider the coefficient of $q^N$. By the way this series is defined, the coefficient of $q^N$ turns out to be the exact number of ways that $N$ can be represented as a square, either zero, one or two respectively as $N$ is zero, a positive square or a non-square.

The coefficients in the $q$-power series expansion of $$\vartheta_3(0,q)$$ count the number of representations of given integers as squares.

``So what?'' you rightfully ask. ``How does this help us?'' After all, we want to count the number of ways an integer can be represented not as a single square, but as the sum of four squares.