arithmetic functions form a ring
Proof. This is essentially a triviality and a little bit of computation.
Many of the ring identities are also obvious. We will prove that is the multiplicative identity, that is commutative and associative, that distributes over , and that the units of the ring are as stated.
To see that is the multiplicative identity, note that
and thus .
To see that is commutative, note that can also be written as
Commutativity is obvious from this of the operation.
Associativity follows similarly. Note that
If one expands similarly, the resulting sum is identical, so the two are equal.
Distributivity follows since
The units of the ring are simply the invertible functions; the entry on convolution inverses for arithmetic functions shows that the invertible functions are those functions with .
|Title||arithmetic functions form a ring|
|Date of creation||2013-03-22 16:30:28|
|Last modified on||2013-03-22 16:30:28|
|Last modified by||rm50 (10146)|