Az Eszterházy Károly Tanárképző Főiskola Tudományos Közleményei. 1998. Sectio Mathematicae. (Acta Academiae Paedagogicae Agriensis : Nova series ; Tom. 25)

PHONG, B. M., Quasi multiplicative functions with congruence property

Quasi multiplicative functions with congruence property BUI MINH PHONG Abstract. We prove that if an integer-valued quasi multiplicative function / satisfies the congruence /(n+p) = /(n) (mod p) for all positive integers n and all primes p^ir, where 7r is a given prime, then f(n) = n

a for some integer a>0. An arithmetical function f(n

) ^ 0 is said to be multiplicative if (n, m) — 1 implies f(nm) - f(n)f(m) and it is called completely multiplicative if this holds for all pairs of positive integers n and m. In the following we denote by M and A4* the set of all integer-valued multiplicative and completely multiplicative functions, respectively. Let N be the set of all positive integers and V be the set of all primes. The problem concerning the characterization of some arithmetical functions by congruence properties was studied by several authors. The first result of this type was found by M. V. Subbarao [7], namely he proved in 1966 that if / E M satisfies (1) f(n -f m) = f(m

) (mod n) for all n, ra E N. then there is an a E N such that (2) f(n) = n

a for aü n G N. A. Iványi [2] extended this result proving that if / E M* and (1) holds for a fixed m E N and for all n E N, then f(n

) has also the same form (2). It is shown in [4] that the result of Subbarao continues to hold if the relation (1) is valid for n E V instead for all positive integers. In [6] we improved the results of Subbarao and Iványi mentioned above by proving that if M E N, / E M satisfy f(M) f 0 and f(n + M) = f{M) (mod n) for all n E N, It was financially supported by OTKA 2153 and T 020295