POLYNOMIAL-ARITHMETIC FUNCTIONS

Abstract

Let A(Ω) denote the ring of arithmetic functions f : Ω → C, where Ω

is the set of monic polynomials over a finite field. Some characterizations

of completely multiplicative arithmetic functions in A(Ω), using distribu

tive property, are established. A necessary and sufficient condition for

the αth power function in A(Ω) to be completely multiplicative is given

for all nonzero real numbers α.