The Gauss-Bonnet Theorem for the noncommutative two torus

In this paper we show that the value at zero of the zeta function of the Laplacian on the non-commutative two torus, endowed with its canonical conformal structure, is independent of the choice of the volume element (Weyl factor) given by a (non-unimodular) state. We had obtained, in the late eighties, in an unpublished computation, a general formula for this value at zero involving modified logarithms of the modular operator of the state. We give here the detailed computation and prove that the result is independent of the Weyl factor as in the classical case, thus proving the analogue of the Gauss-Bonnet theorem for the noncommutative two torus.