# Asymptotics of infinite Gauss sums

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?

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Perhaps you mean a = O(T^(1/2-epsilon))? –  maks Mar 23 '10 at 20:19