In last 30 years singular traces have proved to be a useful tool in different areas of mathematics and its physical applications. Usually the singular traces of operators arising in geometrically or physically inspired problems are computed using the -function residue or a heat kernel asymptotic expansion of these operators (e.g. see [1, 9]). In this paper we study residues of the -function from the classical harmonic analysis point of view. These results extend and complement those in. Throughout the paper (0, ) denotes the space of all (equivalence classes of) real-valued essentially bounded Lebesgue measurable functions on (0, ) equipped with the essential supremum norm.