On the logarithm component in trace defect formulas

In asymptotic expansions of resolvent traces $\Tr(A(P-\lambda)^{-1})$ for classical pseudodifferential operators on closed manifolds, the coefficient $C_0(A,P)$ of $(-\lambda)^{-1}$ is of special interest, since it is the first coefficient containing nonlocal elements from $A$; on the other hand if $A=I$ and $P=D^*D$ it gives part of the index of $D$. $C_0(A,P)$ also equals the zeta function value at 0 when $P$ is invertible. $C_0(A,P)$ is a trace modulo local terms, since $C_0(A,P)-C_0(A,P')$ and $C_0([A,A'],P)$ are local. By use of complex powers $P^s$ (or similar holomorphic families of order $s$), Okikiolu, Kontsevich and Vishik, Melrose and Nistor showed formulas for these trace defects in terms of residues of operators defined from $A$, $A'$, $\log P$ and $\log P'$. The present paper has two purposes: One is to show how the trace defect formulas can be obtained from the resolvents in a simple way without use of the complex powers of $P$ as in the original proofs. We here also give a simple direct proof of a recent residue formula of Scott for $C_0(I,P)$. The other purpose is to establish trace defect residue formulas for operators on manifolds with boundary, where complex powers are not easily accessible; we do this using only resolvents. We also generalize Scott's formula to boundary problems.