A note on some characterizations of arithmetic functions (Q1024960)

The author considers the following extensions of the Dirichlet convolution of arithmetical functions: \[ \begin{aligned} G_{(f\circ g)}(n, m) & = \sum_{d|(n,m)}f(d)g(m/d),\\ G_{(h\circ f\circ g)}(n,m) & = \sum_{d|(n,m)} h(d)f(n/d) g(m/d),\end{aligned} \] where each sum is extended over divisors of the gcd of \(n\), \(m\). As might be expected, these sums are multiplicative when each of \(f\), \(g\), \(h\) is multiplicative. A typical characterization result states that a multiplicative \(f\) is completely multiplicative if and only if \[ G_{(f\circ f)}(n,m)= f(m)\sum_{d|(n,m)} 1. \] When \(n= m\) this yields a known theorem of Carlitz.