Joint distribution in residue classes of polynomial-like multiplicative functions

For any integer-valued arithmetic function, it is reasonable to ask how the values of f are distributed in arithmetic progressions. As stated, this problem is far too general; to get any traction, it is necessary to restrict f . Let us suppose that f is multiplicative and that f is polynomial-like, in the sense that there is a polynomial F (T ) ∈ Z[T ] such that f(p) = F (p) for every prime number p. In this case, Narkiewicz (beginning in [Nar67]) has made a comprehensive study of the distribution of f in coprime residue classes. For a thorough survey of this work, see Chapter V in [Nar84]. See also [Nar12] for a more recent contribution to this subject by the same author.