SCHWEITZER 1975 Problem 4

Prove that the set of rational-valued, multiplicative arithmetical functions and the set of complex rational-valued, multiplicative arithmetical functions form isomorphic groups with the convolution operation $f \circ g$ defined by $${ (f \circ g)(n)= \sum_{d|n} f(d)g(\frac nd}).$$ (We call a complex number $\textit{complex rational}$ , if its real and imaginary parts are both rational.)

B. Csakany