A few months ago I was reading Hardy and Wright’s “An Introduction to the Theory of Numbers” with some friends. While that reading seems to have ended (we made it further than originally expected, to be honest), there was a topic I read about that I decided I wanted to learn more about. When H&W covered Farey sequences, I checked out the Wikipedia page and was fairly delighted to find that there is a statement about Farey sequences that is equivalent to the Riemann hypothesis (RH).
I know nearly nothing about the RH. You take a function defined by an infinite series, extend to the complex plane, notice that the zeroes have to lie in a particular region, and then the claim is that you can narrow down that region even more. Great. Why do I care about the zeroes of this function? Oh, right, because they are related to the distribution of the primes. And how is that again? I have no idea. Probably I should be embarrassed about this, being a math graduate student and not knowing the math behind the most famous open problem.
Well, now seems like as good a time as any to learn more. Farey sequences look fun, and close to some pretty pictures (Ford circles and the ? function). And so my plan is to learn enough about all of that, and the RH, to see if I can understand the relationship between the two. Not so that I can try to prove RH, mind you (I seriously have no delusions about this). It just seems like a fun thing to learn about. Plus maybe it’ll give me an excuse to give a talk in my department’s “graduate seminar”, which I always think is a fun thing to do.
I’ve got a few sources that I plan on looking at (and, indeed, have already been looking at). I found Rademacher’s “Higher Mathematics from an Elementary Point of View” to be quite delightful, and it contains information about the Farey and Ford part of the story. Of course, H&W might also help out here. I’ve got some papers tucked away somewhere I may dig up as well, and when I do, I’ll mention them. And, on the Riemann side of the story, I think I’ll look at Edwards’ “Riemann’s Zeta Function”, which has a section on the connection with Farey sequences.
I figure I’ll try to read at least a little bit every day, and share what I come across here. Perhaps in a month I’ll have some vague understanding (goal: less vague than it is now) about what’s going with the topic. Or perhaps I’ll crap out after a week (or less). But for now, let’s say I’m trying. Starting off a day late (and with an essentially non-mathematical post) means I probably should try to find a day or two and write more than one post. We’ll see. Also, I don’t expect I’ll get close to the 1000 word mark very often. We’ll see.