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From Noam Elkies' AMS article Lattices, Linear Codes, and Invariants, Part I: Elkies has been discussing how difficult it is to find the minimal nonzero length of an element of a lattice $C$.
Sometimes an appropriate response to a difficult mathematical problem is to pose a much harder problem. Here we find the minimal nonzero length intractable, and thus ask for *all* the lengths of vectors of $C$ and their multiplicities. Equivalently, we ask for the following generating function of all the squared lengths, called the *theta function* (or *theta series*) of $C$: $$\Theta_C(z) = \sum_{x \in C} z^{\langle x, x \rangle} = 1 + \sum_{m >0}^{\infty} N_m(C) z^m$$ where $N_m(C)$ is the number of lattice vectors of length $\sqrt{m}$.