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
58 B. M. Phong f(b)f(u) = f(bu) = f(qv + 1) = c (mod q) and f(ab)f(x)f(u) = f(abxu) = f(qT + 1) = C (mod q). These and (5) show that f(x)f(u) 0 (mod q), consequently f(a)f(b) = Cf(ab) (mod q). Hence, we infer from the last relation together and the fact q > |C f(ab) — f(a)f(b)I that Thus, we have proved that (6) holds for all positive integers a and b. By applying (6) with a = b = 1, we have C — 1 and so the proof of Lemma 1 is finished. Lemma 2. Assume that the conditions of the theorem are satisfied. Let Q be a positive integer. Then for each prime divisor q of f(Q) we have q\7rQ. Proof. Let Q be a positive integer and assume on the contrary that there exists a prime q such that q\f(Q) and Í q, 7rQ) — 1. Since (Q,q) = 1, we infer that there are positive integers x and y such that which is a contradiction. Thus the proof of Lemma 2 is finished. Lemma 2 shows that for each prime p, we can write f(p ) as follows: (6) Cf(ab) = f(a)f(b). Qx = qy + 1. By using Lemma 1, it follows from (4) and the fact q ^ 7r that 0 EE f(Q)f(x) = f(Qx) = I(qy + 1) EE 1 (mod 9), \f(p)\=p aip )* 0i p\ consequently (7) l/MI = »", for some non-negative integer a. Now we can prove our theorem.
