How can one express the Dedekind eta function as a sum over the lattice?

Question

The Dedekind eta function $\eta(\tau)$ can be regarded as a formula which assigns a number to a lattice $L \subset \mathbb{C}$. The algorithm is: rotate the lattice so that one of its basis vectors lies along the real axis, then pick another basis vector $\tau$ in the upper half plane, then compute the usual formula $$ \eta(\tau) = q^\frac{1}{24} \prod_{n=1}^\infty (1-q^n) $$ where $q=e^{2 \pi i \tau}$. It would be nice if there were a more "canonical" way to compute it directly from the lattice $L$ (i.e. without this rotate-and-pick-a-basis-vector story which breaks symmetry). I'm looking for a formula similar to that of the Eisenstein series where one sums over all points of the lattice: $$ G_n (L) = \sum_{\omega \in L, \omega \neq 0} \frac{1}{\omega^n} $$ We have the theorem of Jacobi that the 24th power of $\eta$ computes as the discriminant of the lattice, $$ (2\pi)^{12} \eta(\tau)^{24} = 20G_4(\mathbb{Z} + \tau \mathbb{Z})^3 -49 G_6 (\mathbb{Z} + \tau \mathbb{Z})^2, $$ which is great, since it shows that the 24th power of $\eta$ can be defined canonically via a sum over the lattice points... but how about $\eta$ itself?

Answer 1

The functions $G_k$ and $\Delta = \eta^{24}$ can be regarded as functions on the set of lattices because they're modular of level 1. The $\eta$ function isn't modular of level 1 (its level is 24) so there's no natural way to regard it as a function on lattices -- it's a function on lattices with additional "level structure".

Similarly, in order to regularise $G_2$ to get something with good convergence, you end up having to make it depend on some level structure as well.

Answer 2

The logarithmic derivative of the η function is the Eisenstein series G₂, up to elementary factors. The series for G₂ does not quite converge when summed over a lattice, but can be regularized in various ways so that it does converge.

Answer 3

Bruce, the Dedekind function itself is a single sum, $ \prod_{n=1}^{\infty}\left(1-q^n\right)=\sum_{n=-\infty}^{\infty}(-1)^n q^{n (3n+1)/2}$ (Euler). But there is a result about which powers of $\eta$ can be expressed as lattice sums; see for example [Heng Huat Chan, Shaun Cooper, and Pee Choon Toh, The 26th power of Dedekind's $\eta$-function, Adv. Math. 207 (2006), no. 2, 532-–543]. What is your question about?