Cyclotomic Expressions for Representation Functions

Given a subset A of the natural numbers = {0, 1, 2, ···} (resp. of the ring / N of residue classes modulo a positive integer N ), we introduce certain sums of roots of unity associated with A . We study some of their properties, and we use them to obtain new expressions for the classical functions that characterize A , i.e. of the representation function, the counting function and the characteristic function of A . We also give an example of computations of the representation function using such expressions.