A comparative result for multiplicative functions (Q1873207)

Let \(f:\mathbb{N}\to \mathbb{C}\) and \(g:\mathbb{N}\to \mathbb{R}_{\geq 0}\) be multiplicative functions which satisfy \(| f|\leq g\). The authors compare the asymptotic behavior of the sums \(\sum_{n\leq x} f(n)\) and \(\sum_{n\leq x} g(n)\). They generalize celebrated mean-value theorems of E. Wirsing (1967) and G. Halász (1968). The proof is elementary and based on an idea of K.-H. Indlekofer (1993).