A root of unity is a complex number such that for some positive integer . This means that for some integer ; if is relatively prime to , we say that is a primitive -th root of unity, meaning that is not a -th root of unity for any .
Here’s a question: when can we have
if are roots of unity?
This is a little bit vague, in that I did not say what kind of tentative characterization we are looking for. If you were inclined to play devil’s advocate, you could say that equation (1) provides a good enough description. There are, however, less obvious answers that have come in handy in various parts of my research, so let’s look at some of them.Continue reading “Vanishing sums of roots of unity”