  We study extended zeta-function residues on principal ideals of compact
operators and their connections with Dixmier traces. We establish a
Lidskii-type formula for continuous singular traces on these ideals. Using this
formula, we obtain a necessary and sufficient conditions for an arbitrary
operator being Dixmier measurable. These conditions are expressed in terms of
eigenvalues of an operator and an asymptotic of its zeta-function.
