Identity Element for Dirichlet Convolution
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Theorem
Let $f$ be an arithmetic function.
Let $*$ denote Dirichlet convolution.
Let $\iota$ be the identity arithmetic function.
Then:
- $\iota * f = f$
Proof
We have:
\(\ds \map {\paren {\iota * f} } n\) | \(=\) | \(\ds \sum_{d \mathop \divides n} \delta_{d 1} \map f {\frac n d}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \map f n\) |
Hence the result.
$\blacksquare$