Distribution of the values of \(q\)-additive functions on some multiplicative semigroups

Abstract. Let \({\cal P}\) be an infinite subset of primes, $$ \#\{p\leq x~|~p\in {\cal P}\}=(\tau+o(1)){{x}\over {\log x}}\quad(x\to\infty), $$ \({\cal N}\) be the multiplicative semigroup generated by \({\cal P}\). Distribution of the values of \(q\)-additive functions defined on \({\cal N}\) is investigated.