The ring of arithmetical functions in the rational domain

Abstract. The arithmetical functions with rational argument are interpreted as sums of weighted divisors: $$ f(r)=\sum_{d|r}w(d). $$ The logarithmic density of subsets of rational numbers is introduced. It is proved, that if
\( w(d)\geqslant 0\), then asymptotic logarithmic density of the set \(\{r: f(r)\geqslant z\}\) exists.