 Gaussian theta sum

For a full Euclidean lattice $\Lambda$ and a positive definite real symmetric matrix $A$, the Gaussian theta sum is ${\theta }_{\Lambda }(A)={\sum }_{\lambda \in \Lambda }{e}^{-\pi {\lambda }^{T}A\lambda }$. Its absolute convergence follows from Gaussian decay and the polynomial growth of the number of lattice points in a ball. The Poisson summation formula for a Euclidean lattice gives its anisotropic theta functional equation.