Small-parameter asymptotic of a lattice theta sum ($(\det A{)}^{1/2}{\theta }_{\Lambda }(A)\to \mathrm{covol}(\Lambda {)}^{-1}$)

If every eigenvalue of a positive definite symmetric matrix $A$ tends to zero, then the anisotropic theta functional equation and the dominated convergence theorem show that $(\det A{)}^{1/2}{\theta }_{\Lambda }(A)\to \mathrm{covol}(\Lambda {)}^{-1}$. Indeed all eigenvalues of $A^{-1}$ tend to infinity, the zero term of the dual Gaussian theta sum is one, and its remaining terms tend to zero while being dominated by a fixed summable Gaussian. Merely requiring $\det A\to 0$ is insufficient.