The logarithmic residue density of a generalised Laplacian

We show that the residue density of the logarithm of a generalised Laplacian on a closed manifold defines an invariant polynomial valued differential form. We express it in terms of a finite sum of residues of classical pseudodifferential symbols. In the case of the square of a Dirac operator, these formulae provide a pedestrian proof of the Atiyah-Singer formula for a pure Dirac operator in dimension $4$ and for a twisted Dirac operator on a flat space of any dimension. These correspond to special cases of a more general formula by S. Scott and D. Zagier announced in \cite{Sc2} and to appear in \cite{Sc3}. In our approach, which is of perturbative nature, we use either a Campbell-Hausdorff formula derived by Okikiolu or a non commutative Taylor type formula.