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

Multiplicative functions satisfying a congruence property IV. 39 holds for all n,t £ N, p £ V. Applying (14) in the case n = 1 + ip

a 1 and t = 1, one can deduce from (13) that A

k-'f(l + (i + IK1) - A

k~

sf(l + ip

0

11) EE £ j= o - ; (15) = A)s(p«-\l) (mod p), since it is obvious that for a prime p ip Prom (15) we infer that = 0 (mod p) if 1 < j < p a- 1 and so A

kr

s{p

a~\t) = tA

k

fs(p

a-\ 1) (mod p), (16) A;(p°-\p) = Aj(p

a_

1,p) = - - • = A*" V\p) = 0 (mod p) holds for all p £ 'P. By using (14) with k = s and t = p, (16) implies (12). Thus, (12) is proved. Now, from Lemma 2 we have / £ .M* and (17) /(g) = for each q £ P, where a(g) > 0 is an integer. It is clear from (8) that A*/(n+p) = A*/(n) (mod p) for all n £ N and p £ P satisfying the condition p > |yi|. By using (11) in the case Ar = s, we have (18) /(1 + tp) - /(1) - t (/(1 + p) - /(1)) (mod p) for all t £ N and for every prime p > po := max(|A|, A' — 1), because p > k. Considering t = p + 2 and taking account (18) we get Jfc-i = 0 for (/(1 + P) - I)

3 = 0 (mod p),