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Section23.3Making New Functions

Subsection23.3.1First 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.

Definition23.3.1

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 \(n\)

  • \(I(n)=\begin{cases}1& n=1\\0 & n>1\end{cases}\)

In the next computational cell, we define these, as well as a Dirichlet product function. Now let's see what we get!

For instance, what happens if we look for the inverse of \(N\)?

Maybe this is a surprise! But this makes sense, if you remember Example 23.2.3 just previously about \(N = \phi\star u\). Let's confirm that fact numerically as well.

We summarize these explanations as follows.

The second part of Fact 23.3.2 gives an alternate proof for our formula for \(\phi\) from Exercise 9.6.9. \begin{equation*}\phi(n)=N\star\mu(n)=\sum_{d\mid n}N\left(\frac{n}{d}\right)\mu(d)=n\sum_{d\mid n}\frac{\mu(d)}{d}=n\prod_{p\mid n}\left(1-\frac{1}{p}\right)\end{equation*} The last step follows from our initial definition of \(\mu\) in Definition 23.1.1.

Subsection23.3.2More new functions

Next, please try computing the Moebius inversions of our old friends, \(\sigma\) and \(\tau\), 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.

Here one can try this interactively. (You'll need to evaluate the earlier cell after Definition 23.3.1 if you get an error.)

There is loads of fun to be had here. We could try to see what \(\mu\star\mu\) is, or \(u\star u\). Could there be a formula for \(|\mu|\), or could we calculate \(|\mu|\star u\)?

It turns out there are all kinds of other functions you can define. We already saw the first of these informally in our discussion of the Moebius function in Proposition 23.1.4.

Definition23.3.3

If \begin{equation*}n=\prod_{i=1}^k p_i^{e_i}\end{equation*} then we call give the name \(\omega(n)=k\). This is the number of unique prime divisors of an integer. (This is sometimes called \(\nu(n)\) in the literature.)

Definition23.3.4

If \(n=\prod_{i=1}^k p_i^{e_i}\), 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}\, .\end{equation*} This is called Liouville's function.

In both cases, you might want to try a few values to see what these functions look like. See Exercise 23.5.1, or pursue these ideas:

  • What is the value for primes?

  • What is the \(\star\) product of this with something – say, \(u\)?

Finally, although we provide some Sage cells to try things out, you should try them not just with Sage, but also by hand; this is part of the allure of number theory. The sky's the limit. Enjoy!