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

Acta Acad. Paed. Agriensis, Sectio Mathematicae 28 (2001) 35~42 MULTIPLICATIVE FUNCTIONS SATISFYING A CONGRUENCE PROPERTY IV. Bui Minh Phong (Budapest, Hungary) Abstract. It is proved that if an integer-valued completely multiplicative function / with f(n) 0 (Vn E N) and a polynomial P(x

) = Clo + CL^X -f • • • + ükX

k £ Q[x'] satisfy the relation ApP(E)f(n + m) = A

PP{E)f(n) (mod m) for a suitable non-zero integer Ap and for all 71, 171 G N, where P(E)f(n) = a

0f(n) + a

if(n + 1) + • • • + a

kf(n + k), then there is a non-negative integer Oi such that f{Tl) — 1l

a for all n G N. A similar result is true for P(x

) = (x — l)'

1 and a multiplicative function f

. AMS Classification Number: 11A07, 11A25. Keywords: multiplicative functions, congruence properties, characterization of arithmetical functions. 1. Introduction An arithmetical function f(f(n

) ^ 0) is said to be multiplicative if (n,m) = 1 implies f(nm

) = /(n)/(m), and it is called completely multiplicative if this equation holds for all positive integers n and m. Let M and M* be the set of all integer-valued multiplicative and completely multiplicative functions, respectively. Throughout this paper we apply the usual notations, i.e. V denotes the set of primes, N the set of positive intgers and Q the set of rational numbers, respectively. The problem concerning the characterization of some arithmetical functions by congruence properties was studied by several authors. The first result of this