  In this paper we give a proof of the Gauss-Bonnet theorem of Connes and
Tretkoff for noncommutative two tori $\mathbb{T}_{\theta}^2$ equipped with an
arbitrary translation invariant complex structure. More precisely, we show that
for any complex number $\tau$ in the upper half plane, representing the
conformal class of a metric on $\mathbb{T}_{\theta}^2$, and a Weyl factor given
by a positive invertible element $k \in C^{\infty}(\mathbb{T}_{\theta}^2)$, the
value at the origin, $\zeta (0)$, of the spectral zeta function of the
Laplacian $\triangle'$ attached to $(\mathbb{T}_{\theta}^2, \tau, k)$ is
independent of $\tau$ and $k$.
