Carlitz showed necessary and sufficient conditions for an arithmetic function to be a linear combination of two multiplicative functions. He mentions the possibility of generalizing to $k$ multiplicative functions, but as far as I can tell this was never published.

What is known about arithmetic functions which can be represented as linear combinations of some fixed number $k$ of multiplicative functions? Given $k$, how many terms 1, 2, ..., n are needed to either reject it as a linear combination of $k$ multiplicative functions or to generate (partial) multiplicative functions and their coefficients?

[1] L. Carlitz, Sums of arithmetic functions, *Collectanea Mathematica* **20**:2 (1969), pp. 108-126.