On the mean value of multiplicative functions (Q1804186)

The author generalizes the known mean value theorems for multiplicative functions \(f\) of Delange, Elliot, Daboussi and Indlekofer, but not of Heppner. If there exists a constant \(0\leq c<1\) such that \(\sum_{| f(p)- 1| \leq c} {{f(p)-1} \over p}\) converges, \(\limsup_{s\to 1+} \sum_{| f(p)- 1|\leq c} {{| f(p)| -1} \over p} <\infty\) and there exists a nondecreasing function \(A(x)\) satisfying \(\int_ 1^ \infty {{A(x)} \over {x^ 2}} dx< \infty\), \(| \sum_{n\leq x} h(n)|\leq A(x)\), then \(M(f)\) exists. Here \(h\) is defined by \(f* g^{-1}\) with the completely multiplicative function \[ g(p)= \begin{cases} f(p), \quad &\text{if } | f(p)-1 |\leq c\\ 1, &\text{if } | f(p) -1|>c. \end{cases}. \] This theorem is proved by applying Heppner's theorem on the related function \(g\). The paper ends with two interesting examples, that are covered by this theorem, but not by Heppner's (or even Wirsing's).