It, and other pictures like it, express the fact that for a given , if we take the primitive roots for each of the divisors of , together they make up exactly the set of all th roots of unity. The above picture is for the specific case of : the th roots of unity (the dots on the bottom circle) are composed of the primitive roots for , , , , , and (the dots on each of the top circles). I proved this in another post.
Of course, if two sets of complex numbers are the same, then their sums must also be the same. Let’s write for the sum of all the primitive th roots: in my previous post we worked out for certain but weren’t sure how to compute it in other cases. Well, today that’s going to change!
We already know by symmetry that the sum of all the th roots of unity is zero, except when in which case the sum is . Putting all of this together,
That is, the sum of all th roots of unity is the same as summing the primitive roots, , for each divisor of . (The notation means evenly divides , so the summation symbol with underneath means we are summing over all divisors of .)
So what have we gained? Well, we can use this equation “backwards” to compute values for !
- We already know .
- When , the equation tells us that , so it must be that . (Of course we already knew that too.)
- When , we have (note that is not included since is not a divisor of ), so as well.
- , so , which also checks out with our previous knowledge.
- Recall that was the first value we were unsure about. Well, , so must be .
And so on. Since each value of can be computed in this way once we know for all the divisors of (which are smaller than ), we can continue filling in a table of values of like this forever.
For example, to fill in we compute and hence .
Do you notice any patterns? It is not too hard to see that must always be an integer (why?), but so far it has always been either , , or ; will it always be one of those three values? Before my next post you might like to try extending the table of values for further and exploring these questions yourself!