Az Eszterházy Károly Tanárképző Főiskola Tudományos Közleményei. 2001. Sectio Mathematicae (Acta Academiae Paedagogicae Agriensis : Nova series ; Tom. 28)
Bui MINH PHONG, Multiplicative functions satisfying a congruence property IV
38 Bui Minli Phong holds for all n £ N ; t £ N, where A>(Q,<) := A l/( 1 + (Q) - A7(l) («" = 0,1,.. •)• Furthermore , if Q is a •prime, then (9) implies that / \ (11) A}-'(Q,i)= J2 O + J^"^' 1) ( mo d^) /lo/ds for all t £ N, where [a;] denotes the largest integer not exceeding x. This lemma and its proof can be found in [5] (see Lemma 1-2 ). Lemma 2. Let a £ N and f £ M. If (12) /'(n+p a) = /(n) (mod p) for all n £ N and p £ V , then f £ A4* and for each q £ V f(q)=q ai q\ where a(q ) > 0 is an integer. This lemma is indentical to Lemma 3 in [5]. Now we prove Theorem 2. Assume that / £ M and (8) is true for all n, m £ N. First we shall prove that there exists an a £ N such that (12) holds for all n £ N and for all p £ V- If k = 0, then (12) is obviously true. Assume that k > 1 be an integer. Let a be a fixed positive integer such that (13) po ~ max(|A|, jfe - 1) < 2"" 1 Since AA kf{n) = A k{Af(n)), by (8) it follows that A k(Af(n + p a' 1)) = A k{Af(n)) (mod p a_ 1) holds for all n £ N and for all p £ V. Thus, by using Lemma 1 and (13), for s — 1, 2,..., k we have (14) A k~ sf(n + tp"1) - A*"'/(«) - £ ("' ~ 0 A kr S+ j(P a-\i) (mod p ) j—0 ^ J J
