Dirichlet Series of Inverse of Arithmetic Function
Jump to navigation
Jump to search
Theorem
Let $f : \N \to\C$ be an arithmetic function.
Let $g : \N \to \C$ be an Dirichlet inverse of $f$.
Let $F, G$ be their Dirichlet series.
Let $s \in \C$ such that both $\map F s$ and $\map G s$ converge absolutely.
Then $\map F s \cdot \map G s = 1$.
Proof
Let $\varepsilon$ be the identity arithmetic function.
By Dirichlet Series of Identity Arithmetic Function, $\varepsilon$ has Dirichlet series $\map E s = 1$.
By Dirichlet Series of Convolution of Arithmetic Functions, $\map F s \map G s = 1$.
$\blacksquare$