This question has been confusing me and I would love some help.
If $M$ is $n$ by $n$, symmetric, positive definite and integer valued and $n$ is a fixed positive integer, what is the large possible value of
$$\sum_{x \in \mathbb{Z}^n}e^{-\pi^2 {\bf x}^T M^{-1}{\bf x}}\;?$$
For all positive integers $p \ge 1$ consider the matrix $pI$. Then $$\mathbf{x}^T (pI)^{-1}\mathbf{x} = \frac{1}{p} ||\mathbf{x}||^2$$
With a little of algebra one can see that $$e^{- \pi^2\mathbf{x}^T (pI)^{-1}\mathbf{x}} \ge \frac{1}{2}$$ is equivalent to $$||\mathbf{x}|| \le \frac{\sqrt{ \log 2}}{\pi} \sqrt{p}$$
so that taking arbitrarily large $p$ you have an arbitrarily large amount of summands $\ge 1/2$, letting your sum blow up to $\infty$. In particular, a maximum value for that sum cannot be achieved.