= Theta series of a Euclidean lattice
{title2=$\Theta_\Lambda(t)=\sum_{v\in\Lambda}e^{-\pi t|v|^2}$}
= Lattice theta series
{synonym}
The <lattice theta series> of a full-rank <Euclidean lattice> records the multiset of squared lengths of lattice vectors. The Gaussian form shown here converges for $t>0$. For a <flat torus> $\mathbb R^d/\Lambda$, its <heat trace> is $\Theta_{\Lambda^*}(4\pi t)$, by the <spectrum of a flat torus>. Equal theta series do not in general imply orthogonal equivalence of lattices.
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