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

Quasi multiplicative functions with congruence property 57 then there are a positive integer a and a real-valued Dirichlet character x (mod A) such that f(n) = x{n)n

a for all neN,(n,A) = l. in this note we prove this conjecture for a special case, when A = B = 1 and V = A U {x}, where 7r is a fixed prime. Theorem. Let 7r be a given prime and let H (n) be the product of all prime divisors p of n for which p / 7r. If a function f £ QM. and an integer C / 0 satisfy the congruence (4) f(n + 1) = C (mod H(n)) for all n E N, then there is a non-negative integer a such that f(n) - n

a for all n e N. We shall use some lemmas in the proof of our theorem. Lemma 1. Assume that the conditions of the theorem are satisfied. Then f e M*

, i.e f(ab) = f(a)f(b) holds for all a, b £ N. Furthermore C = 1. Proof. Assume that a and b are fixed positive integers. Let q be a prime with the condition (5) q > max(a, 6, |C|, \Cf{ab) - f{a)f(b)\) and q ± TT. Since (ab,q) — 1, one can deduce from Dirichlet's theorem that there are positive integers x,y

:u and v such that ax = qy + 1, (x, ab) = 1, x £ V and bu = qv + 1, (u, abx) = 1, u € V. Then we have abxu — qT + 1, where T := y + v + qyv. Thus, we infer from (4) and the fact / £ QA4

, that f(a)f(x) = f(ax) = f(qy + 1) = C (mod q),