Az Eszterházy Károly Tanárképző Főiskola Tudományos Közleményei. 1998. Sectio Mathematicae. (Acta Academiae Paedagogicae Agriensis : Nova series ; Tom. 25)

PHONG, B. M., Quasi multiplicative functions with congruence property

56 B. M. Phong then (2) holds. Later, in the papers [3]-[5] we obtained some generalizations of this result, namely we have shown that if integers A > 0, B > 0, C / 0, N > 0 with (A, B) = 1 and / E M satisfy the relation f(An + B) = C (mod n) for all n > N

, then there are a positive integer a and a real-valued Dirichlet character x (mod A) such that f(n) = xi

7

1)

7

1" for all n E N, (n,A) = 1. In 1985, Subbarao [8] introduced the concept of weakly multiplicative arithmetic function j(n) (later renamed quasi multiplicative arithmetic functions) as one for which the property f(np) = f(n)f(p) holds for all primes p and positive integers n which are relatively prime to p. In the following let QM denote the set of all integer-valued quasi multiplicative functions. In [1] J. Fabrykowski and M. V. Subbarao proved that if / E QM satisfies (3) f(n + p) = f(n) (mod p) for all n E N and all p E V, then f(n) has the form (2). They also conjectured that this result continues to hold even if the relation (3) is satisfied for an infinity of primes instead of for all primes. This conjecture is still open. Let A C V, and assume that the congruence (3) holds for all n E N and for all p E A. For each positive integer n let H(n) denote the product of all prime divisors p of n for which p E A. It is obvious from the definition that H(n

) j H(mn

) holds for all positive integers n and m, furthermore one can deduce that if / E QM. satisfies the congruence (3) for all n E N and for all p E A

, then f(n -f m) = f(m) (mod H(n

)) for all n, m E N. Thus the conjecture of Fabrykowski and Subbarao is contained in the following Conjecture. Let A, B be fixed positive integers with the condition (yl, B) = 1 and A is an infinite subset of V. If a function f E QM and integer C / 0 satisfy the congruence /(An + B) = C (mod H(n)) for all n E N.