Ring of Arithmetic Functions is Ring with Unity
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Theorem
Let $\AA$ be the set of all arithmetic functions.
Let $*$ denote Dirichlet convolution, and $+$ the pointwise sum of functions.
The ring of arithmetic functions $\struct {\AA, +, *}$ is a commutative ring with unity.
Proof
By Structure Induced by Abelian Group Operation is Abelian Group, $\struct {\AA, +}$ is an abelian group.
By Properties of Dirichlet Convolution, $*$ is commutative, associative and has a unity.
Therefore $\struct {\AA, +, *}$ is a commutative ring with unity.
This needs considerable tedious hard slog to complete it. In particular: link to a proof for distributive To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Finish}} from the code.If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page. |