Time to pull back the curtain a bit! My recent series of posts on complex roots of unity may seem somewhat random and unmotivated so far, but the fact is that I definitely have a destination in mind—we are slowly and inexorably heading towards some really deep and beautiful mathematics. It has just taken a while to build up all the requisite concepts!
Last time, we defined as the sum of all the primitive th roots of unity. We noted that
and used this to deduce values of . In principle we could carry this as far as we want to compute for any , but this doesn’t necessarily give us any insight into the nature of . For example, is always either , , or ? Is there a nicer way to characterize , and/or a quicker way to compute it?
It turns out that is more than just an idle curiosity. In fact, it is famous enough that it has a name: it is known as the Möbius function, and it is usually written . There are indeed nicer ways to characterize and compute . Here’s how it is most commonly defined:
- if has any repeated prime factors, that is, if is divisible by a perfect square.
- Otherwise, if has distinct prime factors, . In other words, if has an even number of distinct prime factors, and if the number of prime factors is odd.
- since is divisible by a perfect square (namely, ).
- since has an odd number of prime factors (namely, one).
- since has two distinct prime factors.
- since has three distinct prime factors.
- since has zero prime factors, and zero is even.
So far, this seems to match up with what we had already deduced about , but it is not at all obvious whether and are the same. Why should adding up a particular set of complex numbers have anything to do with the number of prime factors of ? In my next post, we’ll prove that they are in fact the same. The proof is a nice exercise in combinatorics and relates to Pascal’s Triangle. (Michael Paul Goldenberg hinted at such a proof in a comment on my previous post.)
As you might guess, is deeply related to prime numbers and the Riemann zeta function. It turns out it is also related to the Principle of Inclusion-Exclusion. All this and more to come!