The Ring of Arithmetic Functions of Many Variables

We know various results on arithmetic functions as the Mobius inversion property. And the fact that they form a unique factoriza-tion domain [1] is of much interest to the author. In this note, we develop the theory of arithmetic functions of many variables over algebraic number fields. Let n be a positive number, 01,..., oa rings of algebraic inte-gers in algebraic number fields of finite degrees. Let C be complex numbers, _??_,...,_?? _ n sets of ideals • ‚ 0 of 01,..., on respectively, ƒ¶••ƒ¶n the set of functions of _??_x... x _?? _ to C. For elements a, ƒÀ ofƒ¶, we define the sum and the product as follows: where e. g. (a) (a1,..., an)•¸1 x... x _??_n is of the vector notation. PROPOSITION 1. Under these operations, dl forms an associa-tive commutative ring with the identity. ik1 PROPOSITION 2. ƒ ¶ is an integral domain. PROPOSITION 3. ae1l is invertible if and only if a(1) 0. Of course, (1) is (o„...,on). Let U be the group of units of Ii. If 0*a QQ satisfies a((a)(b))=a(a)a(b), whenever ((a), (b)) =((a1, b1),..., (an, b*))=1, we call a multiplicative, where (a) (b) is (a1b1,•c,anbn) PROPOGITION 4. The set M of multiplicative functions forms a subgroup of U. This is the so-called Mobius inversion. property. We will giv