Mobius Inversion

Suppose we want to find a dirichlet inverse for $1$ since we know that every arithmetic function has an inverse, then we know it exists, and call it $μ$ so we want $μ * 1 = 1 * μ = ϵ$

Dirichlet Inverse for 1

Let

n \in ℕ_{2}

, so that

n = p_{1}^{α_{1}} \dots p_{l}^{α_{l}}

then the dirichlet inverse for 1 is defined as

μ (n) = {\begin{matrix} 0 & if \exists i \in [1, \dots, l] st α_{i} > 1 \\ {(- 1)}^{l} & if \forall i \in [1, \dots, l], α_{i} = 1 \end{matrix}

and

μ (1) = 1

The Dirichlet Inverse for 1 is the Inverse

From the above, note that for any arithmetic function $f$ we have $f = (f * 1) * μ$ .

If an Arithmetic Function Convolved with 1 is Multiplicative so is the Original

Let

f

be arithmetic, then if

f * 1

is multiplicative then so is

f

Since we know that

μ

is multiplicative, then

(f * 1) * μ

is multiplicative but at the same time

(f * 1) * μ = f * ϵ = f

so therefore

f

is multiplicative