Arithmetical functions satisfying a congruence property (Q1262331)

In the paper we consider some problems concerned with characterizing real-valued additive and integer-valued multiplicative functions satisfying some congruence properties. It follows, among others, from our Theorem 1 that if a real-valued additive function f(n) satisfies \(f(An+B)\equiv C (mod n^*)\) for every positive integer n, where \(A\geq 1\), \(B\geq 1\), C are integers and \(n^*\) denotes the product of all distinct prime divisors of n, then \(f(n)=0\) for all n which are prime to A. As a consequence of our Theorem 6 in the paper we have: If integer-valued multiplicative functions f(n) and g(n) satisfy the congruence \(f(An+m)\equiv g(m) (mod n^*)\) for every positive integer n and m, then there are a non-negative integer \(\alpha\) and a real-valued Dirichlet character \(\chi_ A(mod A)\) such that \(f(n)=g(n)=\chi_ A(n)n^{\alpha}\) for all n which are prime to A. Our theorems extend some results of M. V. Subbarao, Bui Minh Phong, K. Kovács and P. V. Chung.