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)

\begin{align} \quad (f * g)(n) = \sum_{d|n} f(d)g \left ( \frac{n}{d} \right ) \end{align}

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)

\begin{align} \quad (\phi * d)(p^k) &= \sum_{a|p^k} \phi(a) d \left ( \frac{p^k}{a} \right ) \\ &= \phi (1) d(p^k) + \phi(p) d(p^{k-1}) + ... + \phi(p^{k-1})d(p) + \phi(p^k)d(1) \\ &= [1][k + 1] + [p - 1][k] + ... + [p^{k-1} - p^{k-2}][2] + [p^k - p^{k-1}][1] \\ &= k + 1 + pk - k + ... + 2p^{k-1} - 2p^{k-2} + p^k - p^{k-1} \\ &= 1 + p + p^2 + ... + p^k \\ &= \sigma(p^k) \end{align}

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