  In this work we study the determinant of the Laplace-Beltrami operator on
rectangular tori of unit area. We will see that the square torus gives the
extremal determinant within this class of tori. The result is established by
studying properties of the Dedekind eta function for special arguments and
refined logarithmic convexity and concavity results of the classical Jacobi
theta functions of one real variable are deeply involved.
