On \(q\)-multiplicative functions taking a fixed value on the set of primes (Q5948333)

Let \(q\geq 2\) be an integer. An arithmetic function \(g\) is called \(q\)-multiplicative if \(g(n)=\prod_jg(\epsilon_jq^j)\), where \(n=\epsilon_0+\epsilon_1q+\cdots+\epsilon_jq^j+\cdots\) is the \(q\)-ary expansion of \(n\). The authors prove the existence of an absolute constant \(C\) with the property that if such a function satisfies \(|g(n)|=1\) for all \(n\), and its value is constant on all sufficiently large primes, then \(g^k(nq)\) holds for all \(n\) with an integer \(k\in[1,C]\).