I would like to evaluate $$ \sum_{q_1 = -\infty}^{\infty} \cdots \sum_{q_N = -\infty}^{\infty} e^{-\sum_{j}\sum_{k} q_{k} A_{kj} q_{j}} $$ with $A$ a real $N\times N$ symmetric matrix.

I know how to compute this when $q$ is continuous (the sum is an integral), and I know how to compute this when $A$ is a scalar (a $1\times 1$, this leads to the Jacobi theta), but If I try to diagonalize $A$, I end up with a transformation for the $q$s that I don't know how to write down in terms of a sum. The transformed $q$s, $q' = O^{T}q$, are linear combinations of the $q$s, and I don't know what the analogous Jacobian-like object would be for summation (in place of integration). Thanks.

  • $\begingroup$ Are your indices right in that expression? $\endgroup$ – Mariano Suárez-Álvarez Dec 13 '15 at 3:01
  • $\begingroup$ @MarianoSuárez-Alvarez I think so, is it misleading that I doubled up on $i$? I can make it a different letter. $\endgroup$ – kηives Dec 13 '15 at 3:03
  • $\begingroup$ @knives The indices are very confusing to me. Does $\prod_i \sum_{q_i}$ actually mean $\sum_{q_1} \ldots \sum_{q_N}$ ? $\endgroup$ – Start wearing purple Dec 21 '15 at 16:24
  • $\begingroup$ @Startwearingpurple Exactly. It is a sum over each of the $q$s. I guess this is causing trouble. I'll try and be more explicit, thanks. $\endgroup$ – kηives Dec 21 '15 at 21:18

This cannot be expressed in terms of elementary (or Jacobi theta) functions: in fact the series $$\Theta\left(\mathbf z | \Omega\right)=\sum_{\mathbf q\in\mathbb Z^N}e^{\pi i \mathbf q\cdot \Omega\cdot \mathbf q+2\pi i \mathbf q \cdot \mathbf z}$$ is a multidimensional generalization of the Jacobi theta function called Riemann theta function. Your case corresponds to setting $\mathbf z=\mathbf 0$, $\Omega=\frac{iA}{\pi}$, i.e. to Riemann theta constants.

For the numerical evaluation, you may have a look at this paper.

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  • $\begingroup$ The eigenvalues of $A$ are necessarily real, but not positive definite, this ruins its applicability doesn't it? $\endgroup$ – kηives Dec 22 '15 at 1:05
  • $\begingroup$ @kηives If $A$ has non-positive eigenvalues, your sum does not converge. $\endgroup$ – Start wearing purple Dec 22 '15 at 15:44
  • $\begingroup$ This is along the lines of what I was looking for, thanks. $\endgroup$ – kηives Dec 24 '15 at 17:09
  • $\begingroup$ Do you know if the Riemann theta has a similar property? mathworld.wolfram.com/JacobisImaginaryTransformation.html $\endgroup$ – kηives Jan 2 '16 at 5:25
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    $\begingroup$ @kηives Yes,,one has something like $\Theta \left( (C\Omega + D )^{-T} \mathbf z | \left( A\Omega +B\right) \left( C\Omega +D \right)^{-1} \right) \sim \Theta (\mathbf z | \mathbf \Omega)$ up to some elementary prefactors for any $2N\times 2N$ matrix $ \left( \begin{array}{cc} A & B \\ C & D \end{array}\right) \in Sp(2N,\mathbb Z )$. The usual Jacobi imaginary transformation corresponds to $N=1$ and $A=D=0$, $B=-C=1$. You can find more details in Mumford's book on theta functions. $\endgroup$ – Start wearing purple Jan 2 '16 at 10:33

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