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. 41 has rational coefficients, consequently Q

i(x*) = (x

s -6{)...(x

s -9l)el

f. Then, by the definition of Ij there is a non-zero integer A

s such that (20) A

sQ

s(E

s)f(n + m) = A

sQ

s(E

s)f(n) (mod m) for all n,m £ N. On the other hand, by using the fact / £ A4*, we have (21) Q

s(E

s)f(sn) = f(s)Q

s(E)f(n). Therefore, (20) and (21) imply that A

sQs(E

s)f [s(n + m)] = A

sQ

s{E

s)f(sn) (mod sm

) (22) AJ{s)Q

s(E)f(n + m) = A

sf(s)Q

s{E)f{n

) (mod sm) for all n, m E N. Since f(s

) ^ 0 and /(s) is an integer, (22) shows that Q

s(x

) £ Ij

. Thus <$(*) = (S(x), Q

s(x)) £ If and so degő(x') = k

, S(x

) = Q

s(x). This implies that for all s £ N, consequently = • • • = 6

k = 1 and S(z) = (x - 1)

A

:

. Thus, Theorem 1 follows directly from Theorem 2. References [1] IvÁNYi, A., On multiplicative functions with congruence property, Ann. Univ. Sei. Budapest, Eötvös, Sect. Math. 15 (1972), 133-137. [2] KÁTAI, I., On arithmetic functions with regularity properties, Acta Sei. Math., 45 (1983), 253-260. [3] ICataI

, I., Multiplicative functions with regularity properties I, Acta Math. Hungar., 42 (1983), 295-308. [4] PHONG, B. M., Multiplicative functions satisfying a congruence property, Studia Sei. Math. Hungar

26 (1991),' 123-128.