On a characterization of some arithmetical multiplicative functions
While a lot is known about regular additive arithmetical functions, the corresponding question for multiplicative functions has been much less investigated. The authors take a step in this direction by showing that if a multiplicative function f satisfies \(\sum_{n\leq x}| f(n+1)- f(n)| =o(g(x))\) but \(\sum_{n\leq x}f(n)\neq o(g(x))\) with a ''nice'' function g, then f is completely multiplicative and \(| f(n)| =n^{\lambda}\) with some constant \(\lambda\geq -1\). To obtain a complete description by showing that \(f(n)=n^ c\) with some complex number c seems to be more difficult and the authors get it only in a very special case. (E. Wirsing has some recent results in this direction.)
distribution of multiplicative functions
scientific article; zbMATH DE number 3891476