# Examples of Dirichlet Convolutions of Two Arithmetic Functions

Recall from The Dirichlet Convolution of Two Arithmetic Functions page that if $f$ and $g$ are two arithmetic functions then the Dirichlet convolution of $f$ and $g$ denoted by $f * g$ is defined to be:

(1)We proved that if $f$ and $g$ are both multiplicative then so is $f * g$. We will now look at some examples of Dirichlet convolutions.

## Example 1

**Show that for $\sigma = \phi * d$.**

Recall that $\sigma(n)$ is the sum of the positive divisors of $n$, $\phi(n)$ is the number of positive integers less than or equal to $n$ that are relatively prime to $n$, and $d(n)$ is the number of positive divisors of $n$. Now observe that since $\phi$ and $d$ are both multiplicative, so is $\phi * d$. So we only need to consider prime powers. Let $p$ be a prime and let $k \in \mathbb{N}$. Then the positive divisors of $p^k$ are $1$, $p$, …, $p^k$. Hence:

(2)Since $\phi * d$ is multiplicative and $\sigma$ and $\phi * d$ agree on prime powers we have that $\sigma = \phi * d$.