convolution inverses for arithmetic functions
If has a convolution inverse , then , where denotes the convolution identity function. Thus, , and it follows that .
Since , we have that . Define .
Now let with and be such that for all with Define
In the entry titled arithmetic functions form a ring, it is proven that convolution is associative and commutative. Thus, is an abelian group under convolution. The set of all multiplicative functions is a subgroup of .
|Title||convolution inverses for arithmetic functions|
|Date of creation||2013-03-22 15:58:32|
|Last modified on||2013-03-22 15:58:32|
|Last modified by||Wkbj79 (1863)|