  Let $P$ be a non-negative self-adjoint Laplace type operator acting on
sections of a hermitian vector bundle over a closed Riemannian manifold. In
this paper we review the close relations between various $P$-related
coefficients such as the mollified spectral counting coefficients, the heat
trace coefficients, the resolvent trace coefficients, the residues of the
spectral zeta function as well as certain Wodzicki residues. We then use the
Wodzicki residue to obtain results about the local counting function of
operators of Dirac and Laplace type. In particular, we express the second term
of the mollified spectral counting function of Dirac type operators in terms of
geometric quantities and characterize those Dirac type operators for which this
coefficient vanishes.
