Theta series of a Euclidean lattice (${\Theta }_{\Lambda }(t)={\sum }_{v\in \Lambda }{e}^{-\pi t|v{|}^2}$)

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 ${\Theta }_{{\Lambda }^{\ast }}(4\pi t)$, by the spectrum of a flat torus. Equal theta series do not in general imply orthogonal equivalence of lattices.