Completely q - multiplicative functions : the Mellin transform approach

(1.1) f(aq + b) = f(aq) + f(b) or f(aq + b) = f(aq)f(b) for all r ≥ 0, a ≥ 0 and 0 ≤ b < q. Note that these equations force f(0) = 0 or f(0) = 1 respectively. These functions are called q-additive and q-multiplicative, respectively. It is easy to see that the functional equations imply that these functions are defined for all integers, when the values f(aq) are known for 1 ≤ a ≤ q − 1 and all r ≥ 0. In [MM83] the case of q-additive functions is studied with the help of the Dirichlet generating function of f(n) under the additional assumption that ∑∞ k=0 f(aq )z is rational. The “typical” result for the summatory function ∑ n<N f(n) is a (finite) linear combination of terms N (logN)φ(logqN), where k ≥ 0 is an integer and φ is a continuous periodic function of period 1. A well known example in this context is the binary sum-ofdigits function s(n) for which Delange’s summation formula (cf. [De75]) holds: ∑