I'm looking for a description of the error term in the asymptotics of
$\sum_{k\in \mathbb{Z}^n} \exp(-(|k|^2+|k+a|^2)/(2 T))$
as $T \to \infty$, which should be uniform in the parameter $a \in \mathbb{Z}^n$. Particularly interesting is the case of large $a=O(T^{1-\epsilon})$ for some small $\epsilon$. Does anyone know of results in this direction?
