Fourier / CONCEPT

MatrixGaussianPoisson

For every natural rank r, real positive definite A and complex b with positive real part, sum_a exp(-pi b Q_A(a)) equals sqrt(det A)^(-1) b^(-r/2) sum_k exp(-pi Q_(A^(-1))(k)/b), with a and k ranging over all integer vectors of rank r. Both primal and inverse-form term norms are summable. The power is principal. The proof uses coordinate Poisson summation, the genuine matrix Fourier integral and absolute convergence of both lattice sums. No integer-coordinate diagonalization is used, and rank zero remains in scope.