Section 23.3 Making New Functions
Subsection 23.3.1 First new functions
In order to see what good this does, let's see what happens when we mess around and make Dirichlet products with functions we know. We already know two of these functions, and I give you a third.Definition 23.3.1.
We define a new simple arithmetic function to go along with those from Definition 19.2.9.
\(u(n)=1\) for all \(n\)
\(N(n)=n\) for all \(n\)
\(\displaystyle I(n)=\begin{cases}1& n=1\\0 & n>1\end{cases}\)
Fact 23.3.2.
We may identify the following Dirichlet products as known functions.
\(\displaystyle \phi\star u=N\)
\(\displaystyle N\star\mu=\phi\)
\begin{equation*}
\phi(n)=\left(N\star\mu\right)(n)=\sum_{d\mid n}N\left(d\right)\mu\left(\frac{n}{d}\right)=
\end{equation*}
\begin{equation*}
\sum_{e\mid n}N\left(\frac{n}{e}\right)\mu(e)=n\sum_{e\mid n}\frac{\mu(e)}{e}=n\prod_{p\mid n}\left(1-\frac{1}{p}\right)\text{.}
\end{equation*}
The middle step follows if we let \(e=n/d\text{,}\) since that sum will also go through all divisors of \(n\text{.}\) The last step follows from our initial definition of \(\mu\) in Definition 23.1.1.Subsection 23.3.2 More new functions
Next, please try computing the Moebius inversions of our old friends, \(\sigma\) and \(\tau\text{,}\) by hand for several values. (Hint: try primes and perfect powers first, as they don't have many divisors!) You can try something out here in Sage as well. If you are online, in the next few cells one can try this interactively. (If you get an error, you'll need to evaluate the earlier cell after Definition 23.3.1.) There is a load of fun to be had here. We could try to see what \(\mu\star\mu\) is, or \(u\star u\text{.}\) Could there be a formula for \(|\mu|\text{,}\) or could we calculate \(|\mu|\star u\text{?}\) It turns out you can define all kinds of other functions. We already saw the first of these informally in our discussion of the Moebius function in Proposition 23.1.5.Definition 23.3.3.
If
\begin{equation*}
n=\prod_{i=1}^k p_i^{e_i}
\end{equation*}
then we can give the name \(\omega(n)=k\) to the number of unique prime divisors of an integer. (This is sometimes called \(\nu(n)\) in the literature.)
Definition 23.3.4.
If \(n=\prod_{i=1}^k p_i^{e_i}\text{,}\) we summarize the parity of the total powers of primes dividing a number as
\begin{equation*}
\lambda(n)=(-1)^{e_1+e_2+\cdots+e_k}\text{.}
\end{equation*}
This is called Liouville's function.
What is the value for primes?
What is the \(\star\) product of this with something – say, \(u\text{?}\)