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 59 Proof of the theorem. We shall prove that f(n

) = n

a is satisfied for all n £ N, where a > 0 is given in (7). Let n, s be positive integers. By (4), we have f(n7T

2

s) = f ((mr

2

s - 1) + l) = 1 (mod H(nir

2

s - 1)). On the other hand, it follows from Lemma 1 and (7) that n

af(n7T

2

s) = n

af(n)7T

2a

s = f(n)(mr

2

s)

a = f{n) (mod II{nir

2

s - 1)). These imply f(n) = n

Q (mod fí

(nit

2

s - 1)), therefore, by setting s —

> oo, we have H(nir

2

s — 1) —> oo and so f(n) = n

a

. This holds for each positive integer n, consequently it also holds for all ÍIÉ

N

. The theorem is proved. References [1] J. FABRYKOWSKI and M. V. SUBBARAO, A class of arithmetic functions satisfying a congruence property, Journa.1 Madras University, Section B 51 (1988), 48-51. [2] A. IvÁNYl, On multiplicative functions with congruence property, Ann. Univ. Sei. Budapest, Sect. Math. 15 (1972), 133-137. [3] I. Joó and 8. M. PHONG, Arithmetical functions with congruence properties, Ann. Univ. Sei. Budapest, Sect. Math. 35 (1992), 151-155. [4] B. M. PHONG, Multiplicative functions satisfying a congruence property, Periodica Math. Hungar. 26 (1991), 123-128. [5] B. M. PHONG, Multiplicative functions satisfying a congruence property V, Acta Math. Hungar. 62 (1993), 81-87. [6] B. M. PHONG and J. FEHÉR

, Note on multiplicative functions satisfying congruence property, Ann. Univ. Sei. Budapest, Sect. Math. 33 (1990), 261-265. [7] M. V. SUBBARAO, Arithmetic functions satisfying congruence property, Cariad. Math. Bull., 9 (1966), 143-146. [8] M. V. SUBBARAO, Amer. Math. Soc. Abstract, 86 # T-LL-15, p. 324. Bui MINH PHONG EÖTVÖS LORÁND UNIVERSITY DEPARTMENT OF COMPUTER ALGEBRA PÁZMÁNY PÉTER SÉT. 2. INF. ÉP. H-1117 BUDAPEST, HUNGARY E-mail: buiicompalg.elte.hu