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 three new simple arithmetic functions.
  • \(u(n)=1\) 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 our comment above about \(N = \phi\star u\). Let's confirm that fact numerically as well.

We summarize this as follows.

Sticking with \(\phi\) for a little longer, notice that this gives an alternate proof of our formula for it, based on our first definition of \(\mu\). \[\phi(n)=N\star\mu(n)=\sum_{d\mid n}N\left(\frac{n}{d}\right)\mu(d)=n\sum_{d\mid n}\frac{\mu(d)}{d}=n\prod_{p\mid n}\left(1-\frac{1}{p}\right)\]

Subsection23.3.2More new functions

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.

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 \(|\mu|\star u\)?

And it turns out there are all kinds of other functions you can define.

Definition23.3.3
If \[n=\prod_{i=1}^k p_i^{e_i}\] then we call \(\omega(n)=k\). 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 \[\lambda(n)=(-1)^{e_1+e_2+\cdots+e_k}\, .\] This is called Liouville's function.

In both cases, you might want to try a few values to see what this looks like. What is it for primes, for instance?

Then, what is the \(\star\) product of this with something, or where else does it show up? You might want to try the \(\star\) product with \(u\).

Finally, 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!