The theta function of a lattice is defined to be
$$ \vartheta_\Lambda = \sum_{v\in\Lambda} q^{{\Vert v\Vert}^2}$$
which yields as a coefficient of *q ^{k}* the number of vectors of norm-squared

*k*.

On the other hand, the Jacobi theta function is given by $$ \vartheta(u,q) = \sum_{n=-\infty}^\infty u^{2n}q^{n^2}$$ and we have the obvious fact that if $\Lambda = \mathbb{Z}$ with its usual intersection form, then $\vartheta(1,q)$ is the theta function for that lattice.

We also have the fact that $\vartheta_{\Lambda_1 \oplus \Lambda_2} = \vartheta_{\Lambda_1}\vartheta_{\Lambda_2}$, and so we can decompose our theta functions into products of theta functions of primitive lattices.

Combining these facts, it is not entirely ridiculous to hope that there is some way to write, for a lattice of rank *k*, a ``theta function'' of the form
$$\vartheta(u_1, \ldots, u_k, q)$$
such that $\vartheta(1, \ldots, 1, q)$ is the ordinary theta function of the lattice. In some sense, the *u*-variables keep track of the basis elements of the lattice which immediately raises the question as to well-definedness of such an idea; it is worth noting that for the lattice $\Lambda = \bigoplus_i\mathbb{Z}$ that this definition does make sense.

So is there any literature on such objects? Do they make sense for lattices which are not just sums of copies of ℤ? Do they have nice relations akin to those of normal theta functions?