On some arithmetical multiplicative functions

Abstract

We characterize some non-negative multiplicative functions $f(n)$ such that $\lim_{x\rightarrow +\infty}\frac{1}{x}\sum_{{1\leq n\leq x }\atop {n\in A }} f(n)$ exists and is positive, but there exists a subset $A(f)$ of $N$ of density $1$ such that $\lim_{x\rightarrow +\infty }\frac{1}{x}\sum_{{1\leq n\leq x }\atop {n\in A(f)}} f(n)=0$. An application to the case of the Ramanujan $\tau$-function is provided.