Acta Mathematica Academiae Scientiarum Hungaricae 63. (1994)
1994 / 1. szám - Joó I.: Arithmetic functions satisfying a congruence property
Acta Math. Hungar. 63 (1) (1994), 1-21. ARITHMETIC FUNCTIONS SATISFYING A CONGRUENCE PROPERTY I. JOÓ (Budapest) 1. An arithmetic function /(n) is multiplicative (resp. additive), if /(nm) = f(n)f(m) (resp. f{nm) - f(n) + /(m)) for any pair n,m of relatively prime positive integers, and completely multiplicative (resp. completely additive), if the above equality holds for any pair n,m. The problem concerning the characterization of an integer-valued power function as an integer-valued multiplicative function satisfying a congruence property was studied by several authors. In 1966, M. V. Subbarao [17] proved that if an integer-valued multiplicative function f{n) satisfies the congruence (1.1) f(n + m) = f(m) (mod n) for every positive integer n and m, then there is a non-negative integer a such that (1.2) f(n) = na (n= 1,2,...). In [3], A. Iványi extended this result proving that if an integer-valued completely multiplicative function /(n) satisfies (1.1) for a fixed positive integer m and for every positive integer n, then /(n) is also of the same form (1.2). Furthermore A. Iványi also showed that the same assertion can be deduced from the congruence (1.3) f(n +m) = f(n) +f(m) (mod n) instead of (1.1) for an integer-valued multiplicative function /(n) and for every positive integer n,m. In the space of sequences {xn} we define the operators E,I and Д as follows: Ixn .— Xn) Exn .— 3:^+1 and Axn .— xr/. If P(x) = a0 -f a\X a,kxk is an arbitrary polynomial with integer coefficients, then we extend the above definition as follows: E(E^xn .— üQXn -f- d\xn+\ + ...T öfc%п-\-к*