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. 37 (6) P(x) = x

M - 1 (M e N), A = N, B = V. Hence we apply the method of I. Kátai [2]-[3] to prove the following. Theorem 1. Let f 6 .A4* with condition (5) f(n) ^ 0 for all n £ N. Let P{x) be a non-zero -polynomial with rational coefficients for which there exists a suitable non-zero integer Ap such that (6) ApP(E)f{n + 777.) = A

PP(E)f(n) (mod m) for all Ii E N and m £ N. Then there is a non-negative integer a such that (7) f(n) = n

a for all n £ N. We mention that in the special case P(x) = The orem 1 is true under the assumption / £ M.. Theorem 2. Let f G M and let A ^ 0, k > 0 be integers. If A

h f(n) satisfies the relation (8) AA

kf(n + m) = AA

k f(n) (mod m) /or all n 6 N and m G N. i/ieri (7) holds. 2. Proof of Theorem 2 In the proof of Theorem 2 we shall use the following results. Lemma 1. Let f(n) be an integer-valued arithmetic function and let k G N, Q e N. If A

kf(n) satisfies the relation (9) A

kf(n + Q) = A

kf(n) (mod Q) for all 77. £ N, then for s = 1,2,... ,k (10) A

k~

sf(n+tQ) - A

k~'f(n) = £ (" ~ ^ A

k~

s+

j (Q, t) (mod Q) j=o ^ J '