Suppose one asks what fraction of matrices over a fixed finite field are invertible. For nxn matrices, one can calculate this easily. In order to make sense of this, one might be tempted to take the limit with n. As it happens, the limit converges. Not only that, it converges to a very nice function (of the size of the field). It's a theta function. It shows up all over the place in number theory. It's modular. Its Mellin transform is closely related to the Riemann zeta function.
Now why does this function show up here?