Kevin Knudson: Welcome to My Favorite Theorem, a math podcast and so much more. I'm Kevin Knudson, professor of mathematics at the University of Florida. Here is your other host.

Evelyn Lamb: Hi, I'm Evelyn Lamb. I'm a math and science writer in Salt Lake City, Utah. I have left the county two times since this all happened. We don't have a car, so when I leave my home, it is either on feet or bicycle, which is your feet moving in a different way. But I have biked out of our county now into two different other counties. So it's very exciting.

KK: Fantastic. Well, I do have a car. I bought gas yesterday for the first time since May 26, I think. And yesterday was June 30.

EL: Yes.

KK: And I've gotten two haircuts, but it looks like you've gotten none.

EL: Yes. That’s correct. I’m probably the shaggiest. I've been in a while. My I normally this time of year is buzzcut city, which I do at home anyway. But I don't know.

KK: I will say I’m letting it get a little longer actually. I know I said I got a haircut, but you know, Ellen likes it longer somehow. So here we go. This is where we are. My son's been home for three months, and we haven't killed each other. It's all right.

EL: Great. Yeah, everything's doing as well as can be expected, I suppose. If you're listening to this in the future, and somehow, everything is under control by the time we publish this, which seems unlikely, we are recording this during the 2020 COVID-19 pandemic, right, which—I guess it still stays COVID-19 even though it's 2020 now, to represent the way time has not moved forward.

KK: Right. Time has no meaning. And you know, Florida now is of course becoming a real hotspot, and cases are spiking. And I'm just staying home and, and I have four brands of gin, so I'm okay.

EL: Yeah. Anyway!

KK: Anyway, let's talk math. So we're pleased today to welcome Daniel Litt. Daniel, would you please introduce yourself?

Daniel Litt: Hey, thank you so much. It's really nice to be here. I'm Daniel Litt. I'm an assistant professor at the University of Georgia in Athens, Georgia, likewise, a COVID-19 hotspot. I also have not gotten gas, but I think I've beat your record, Kevin. I haven't gotten gas since the pandemic began.

KK: Wow. That’s pretty remarkable.

DL: I’ve driven, maybe the farthest away I've driven from home is about a 15-minute drive, but those are few and far between.

KK: Sure.

DL: So yeah, I'm really excited to be here and talk about math with both of you.

KK: Cool. All right. So I mean, this podcast is—actually, let’s talk about you first. So you just moved to Athens, correct?

DL: I started a year ago.

KK: A year ago, okay. But you just bought your house.

DL: That’s right. Yeah. So I actually live in northeast Atlanta, because my wife works at the CDC, which is a pretty cool place to work right now.

KK: Oh!

EL: Oh wow.

KK: All right. Is she an epidemiologist?

DL: She does evaluation science, so at least part of what she was doing was seeing how the CDC’s interventions and deployers, how effective they were being help them to understand that.

KK: Very cool. Well, now it would be an interesting time to work there. I'm sure it's always interesting, but especially now. Yeah. All right. Cool. All right. So this podcast is called my favorite theorem. And you've told us what it is, but we can't wait for you to tell our listeners. So what is your favorite theorem?

DL: Yeah, so my favorite theorem is Dirichlet’s theorem on primes in arithmetic progressions. So maybe let me explain what that says.

KK: Please do.

EL: Yes, that would be great.

DL: Yeah. So a prime number is a positive integer, like 1, 2, 3, 4, etc, which is only divisible by one and by itself. So 2 is a prime, 3 is a prime, 5 is a prime, 7, 11, etc. Twelve is not a prime because it's 3 times 4. And part of what Dirichlet’s theorem on primes in arithmetic progressions tries to answer, part of the question it answered, is how are primes distributed? So there is a general principle of mathematics that says that if you have a bunch of objects, they're usually distributed in as random a way as possible. And Dirichlet’s theorem is one way of capturing that for primes. So it says if you look at an arithmetic progressions—that’s, like 2, 5, 8, 11, 14, etc. So there I started at 2 and I increased by 3 every time. Another example would be 3, 6,9, 12, 15, etc—there I started at 3 and increased by 3 every time. So Dirichlet’s theorem says that if you have one of those arithmetic progressions, and it's possible for infinitely many primes to show up in it, then they do. So let me give you an example. So for 3, 6, 9, 12, etc, all of those numbers are divisible by 3. So it's only possible for one prime to show up there, namely 3.

EL: Right.

DL: But if you have an arithmetic progression, so a bunch of numbers which differ by all the same amount, and they're not all divisible by some single number, then Dirichlet’s theorem tells you that there are infinitely many primes in that sequence. So for example, in the sequence 2, 5, 8, 11, etc, there are infinitely many primes, 5 and 11 being the first two [editor’s note: the first primes after 2. But it’s just odd for an even number to be prime]. And it tells you something about the distribution of those primes, which maybe I won't get into, but just their bare existence is really an amazing theorem and incredible feat of mathematics.

EL: So this theorem, I guess, for some of our listeners, and for me, it probably sort of reminds them in some ways of like twin primes or something, these other questions about distributions of primes. Of course, twin primes, you don't need a whole arithmetic progression, you just need two of them. That would be primes that are separated by two, which other than 2 and 3 is the smallest gap that primes can have. And, of course, twin primes is not solved yet.

DL: Yeah, we don’t know that there are infinitely many.

EL: Yeah, people think there are but you know, who knows? We might have found the last one already. I guess that's unlikely. But Dirichlet was proved a long time ago. So can you give me a sense for why this is a lot easier than twin primes?

DL: Yeah, so part of the reason, I think, is that twin primes are much sparser than primes in any given arithmetic progression. So just to give you an example, if you have a bunch of numbers, one way of measuring how big they are is you could take the sum of 1 over those numbers. So for example, the sum of 1/n, where n ranges over all positive integers, diverges; that sum goes to infinity. And the same is actually true for the primes in any fixed arithmetic progression. So if you take all the primes in the sequence 2, 5, 8, 11, etc, and take the sum of one over them, that goes to infinity, since there's a lot of them. On the other hand, we know that if you do the same thing for twin primes, that sum converges to a finite number. And that number is pretty small, actually. We know, up to quite a lot of accuracy, what it looks like. And that already tells you that they're sort of hard to find. And if you have things that are hard to find, it's going to be harder to show that there are infinitely many of them. I mention this sum of reciprocals point of view because it's actually crucial to the way Dirichlet’s theorem is proven. So when you prove Dirichlet’s theorem, it's one of the these really amazing examples where you have a theorem that's about pure algebra. And you end up proving it using analysis. So in this case, the theory of Dirichlet L-functions. And understanding that sum of reciprocals is kind of key to understanding the analytic behavior of some of these L-functions, or at least it’s very closely related.

KK: So I didn't know that result about the reciprocals of the twin primes converging. So even though we don't know that there are infinitely many, somehow…

DL: Yeah, in fact, if there are finitely many then definitely that sum would converge, right?

KK: Yeah, right. That’s—and we even know an estimate of what the answer is? Okay. That’s fascinating.

DL: Yeah, and what you have to do to prove that is show that these primes are sufficiently sparse. And then and then you win. EL: So once again, I am super not a number theorist. So I'm just going to bumble my way in here. But to me, if I'm trying to show that something diverges, I show that it's sort of like 1/n, and if it converges, it's sort of like 1/n2 or, or worse, or better, or however, you want to morally rank these things. So I guess I could imagine it not being that hard to show that twin primes are sort of bounded by n2, or you're like bounded by 1/n2 squared, the reciprocals of that, would that be a way to do this? Or am I totally off?

DL: It’s something like that. You want to show they're very spread out. Yeah, with primes, I do want to mention, so you mentioned like you want to say something like between 1/n or 1/n2. So primes are much, much rarer than integers, right? So it's really somewhere between those two.

EL: Yeah.

DL: So for example, understanding the growth rate of those numbers—the growth rate of the primes and the growth rate of the primes in a given arithmetic progression—is pretty hard. Like that's the prime number theorem, it’s one of the biggest accomplishments of 19th-century mathematics.

KK: Right. Does that help you prove that, though? Maybe it does, right? Maybe not?

DL: Yeah, so proving that the sum of the reciprocals of the primes diverges is much, much easier than the prime number theorem. And as you can prove that in, like, a page or page and a half or something. But it's very closely related to the key input of the prime number theorem, which is that the Riemann zeta function, the subject of the Riemann hypothesis, has a pole at s=1.

KK: All right. Okay. So what's so compelling about this theorem for you?

DL: Yeah, so what I love about it is that it's maybe one of the earliest places, aside from the prime number theorem itself, where you see some really deep interactions between algebra and complex analysis. So the tools you bring in are these Dirichlet L-functions, which are kind of generalizations of the Riemann zeta function. And they're really mysterious and awesome objects. But for me, what I find really exciting about it is that it's like the classic oldie. And people have been kind of remaking it over and over again for the last, like, century. So there's now tons of different versions of the Dirichlet theorem on primes in arithmetic progressions in all kinds of different settings. So here's an example. In geometry, you have a Riemannian manifold, which is kind of a manifold with a notion of distance on it. There's a version of Dirichlet’s theorem for loops in a Riemannian manifold, the first cases of which are maybe do that Peter Sarnak in his thesis. There are versions for over function fields. So I'm not going to be precise about what that means, but if you have some kind of geometric object that's kind of like the integers, you can understand it well and understand the behavior of primes and that kind of object, and how they behave in something analogous to an arithmetic progression. There's something called the Chebotarev density theorem, which tells you if you have a polynomial, and you take the remainder of that polynomial when you divide by a prime, how does its factorization behave as you vary the prime? So there's all kinds of versions of it, and it's a really exciting and cool sort of theme in mathematics.

EL: So kind of getting back to the the more tangible number theory thing—which I guess it's kind of funny that we think of numbers as more tangible when they're sort of the first example of an incredibly abstract concept. But anyway, we'll pretend numbers are tangible. So how does this relate, I remember, and I don't even remember now, I must have been writing some article that related to this, but looking at your primes that are your 1 more than a multiple of 6 versus 1 less and looking at whether there are more or fewer of these. So these are two different arithmetic progressions. The one that's like, you know, 7, 13, let's see if I can add by 6, 19, this, that progression, versus the 5, 11, etc, progression. So is this related to looking at whether there are more of the ones that are one more one less or things like that?

DL: For sure.

EL: I feel like there are all these interesting results about these biases and the distributions.

DL: Yeah, so people call this prime number races.

EL: Yeah.

DL: So what you might do is you might take two different arithmetic progressions and ask are there more prime numbers, like, less than a billion, say, in one of those progressions as opposed to the other? And there are actually pretty surprising properties of those races that I think are not totally well understood. So like even even this recent work of Kannan Soundararajan and Robert Lemke Oliver on this kind of thing.

EL: Oh, yeah, that’s what I was writing about!

DL: Which, yeah, shows some sort of surprising biases. And so that's the reason people think those are cool, is exactly this principle I mentioned before, this general principle of math that things should be as random as they can be. And there are maybe some ways in which our random models of the primes are not always totally accurate. And so understanding the ways in which they're inaccurate and how to fix that inaccuracy, like how to come up with a better model of the primes, is a really big part of modern number theory.

EL: But I guess, the Dirichlet theorem is what you need before you start looking at any of these other things, is you need to know that you can even look at these sequences.

DL: Right. Exactly. Yeah. I mean, how do you study the statistics of a sequence you don't know is infinite? Yeah.

EL: Right.

DL: One thing I’ll mentioned, one cool thing about it is it lets you—it’s not just an abstract existence result. Like, sometimes you just need a prime which is, like, 7 mod 23 to do some mathematical computation. Okay, and if it's 7 mod 23, then it's pretty easy to find one. You can take 7. But if you need a prime, that's a mod b, its remainder upon division by b is a, it's sort of hard to make one in general. And the fact that Dirichlet’s theorem gives them to you is actually really useful. So at least for a mathematician who cares about primes, it's something that just comes up a lot in daily life.

KK: But it's not constructive, though.

DL: Yeah, that's, that's right. It does kind of guarantee that there will be one less than some explicit constant, so in some sense, it's constructive, but it doesn’t, like, hand one to you.

EL: But still, I guess a lot of the time, you probably don't actually need a particular one. You just kind of need to know that there is one.

DL: Yeah.

EL: And where did you first encounter this theorem?

DL: I guess it was, I was probably reading Apostol’s number theory book when I was in college. But I think for me, I didn't really grok it until some other more modern version of it, like one of these remakes showed up for me in my own work. So I wanted to make a certain construction of algebraic curves. So that's some kind of geometric objects defined by some polynomial equations, which have some special properties. And it turned out that for me, the easiest way to do that was to use some version of Dirichlet’s theorem in some kind of geometric context.

KK: Very cool.

DL: So that was really exciting.

KK: Yeah. Well, it's it's nice when, like you say, when the oldies come up on your jukebox. They're useful.

DL: Yeah, exactly.

KK: So another fun thing about this podcast is that we ask our guests to pair their theorem with something. And I mean, I think Evelyn and I are just dying to know what pairs well with Dirichlet’s theorem on primes in arithmetic progressions.

DL: So for me, it's the Arthur Conan Doyle stories about Sherlock Holmes.

KK: Okay.

DL: For a couple different reasons. So first of all, because he's all about making connections between these sort of seemingly unrelated things, just like Dirichlet’s theorem is about making connections between, somehow for the proof, it's about connecting these things in algebra, primes, to things in complex analysis, these L-functions, but then also because it's an oldie that's been remade over and over again. It's still constantly being remade, like with the new BBC Sherlock show.

KK: It’s the best. Yeah, I remember when that was coming out. My wife and I were just so excited every time a new season come out, you know, just “Sherlock! Yes!”

DL: Yeah, just like I'm so excited every time a new version of Dirichlet’s theorem on primes in arithmetic progression comes out.

EL: Yeah, I haven't watched any of the Sherlock TV or movies yet. But we're watching a little more TV these days, and that might be a good one for us to go look at.

KK: It is so good. I mean, the first episode…

EL: Is that the one with Benedict Cumberbatch?

KK: Yeah, but the first one, just, I mean, it just grabs you. You can't not watch it after that. It's really, really well done.

DL: Yeah, they're really fun. Although—oh, go on.

KK: I was going to say the last one, the very last episode, I thought was a bit much.

DL: I don't know that I watched the last season.

KK: Yeah, it was a little…yeah. But you know, still good.

DL: I was reading a couple of the old short stories in preparation for this podcast. Those are also, I highly recommend.

KK: Which ones did you read?

DL: My favorite one that I read recently was, I think it's called the Adventure of the Speckled Band.

KK: Mm hmm.

EL: Oh, yeah.

DL: It's one of the classics.

KK: Right. Yeah. And I think they based one of the episodes on that one, too.

DL: Yeah. that’s right. Yeah.

EL: Yeah, that's a good one. I haven't read all of the Sherlock Holmes it seems like they're practically infinitely many of them. But you know, I had this collection on my Nook and we were moving, so it was like light, and I could read it in the hotel room easily and stuff. And as we were moving to Utah, I think the very first Sherlock Holmes one is set in Utah, or like part of it is set in Utah.

DL: Yeah, maybe the Sign of Four?

EL: Yes, I think it’s the Sign of Four.

DL: Yeah, I think it's one of the first two novellas. So I’ve read every single Sherlock Holmes when I was when I was in high school or something.

EL: Okay. But I was just like, of all things. I didn't know, I hadn't ever read any Sherlock Holmes before. And, like, this British guy writing about this British detective, and it’s set in the state I’m about to move to. It just seemed incredibly improbable to me.

DL: Yeah, I guess he had some kind of fascination with the U.S. because there's that one, which is sort of set in Utah as it was being settled, I guess.

EL: Yeah.

DL: And then there's the case of the five orange pips or something, which actually in a timely way crucially involves the KKK. And so yeah, so there's a lot of sort of interesting interactions with American history.

EL: Yeah, I don't I don't remember if I've read the orange pips.

KK: That figures in the TV series too.

EL: Okay. Yeah, I kind of forgot about those. Those might be a fun thing to go back to, since unlike you, I have not read all of them, and there always seem to be more that I could kind of dive into. I think I kind of tried to read too many at one time, and I just got fed up with what a jerk he is. Self righteous, smug guy.

DL: Yeah, definitely.

EL: Which doesn't make it not entertaining.

DL: If you like this stuff, there's a nother thing I was thinking of pairing. pairing with the theorem. There's a novel by Michael Chabon about a sort of very elderly Sherlock Holmes. Which I don't quite remember the name but part of it is about, you know, what it's like to be Sherlock Holmes when you're 90 and all your friends have left you, and so maybe that might, might appeal to you if you find him sort of an annoying character.

EL: Yeah. Could that be the Yiddish Policeman's Union?

DL: I don't think so. It's a much shorter book.

EL: Okay. That’s the title I could remember.

DL: That one is also excellent. It just doesn't have Sherlock Holmes in it. [Editor’s note: the book is The Final Solution: A Story of Detection.]

EL: Okay. Well, when you were talking earlier about the theorem, you used the word, I think you used the word remake or sequel or something. So I was wondering if you were going to pick movies, or something like that for your pairing. But this kind of works, too, because each one, it’s not a not remakes exactly—I guess with the movies there are remakes, movies and TV shows. But the stories are all, like, some new sequel. Like, here's a slightly different adventure that Sherlock goes on. And slightly different clues that he finds.

DL: Yeah, exactly. That's one thing that I love about math in general is that so much of it is you look at something classic, and then you put a little spin on it. Like I do a little exercise with some of the grad students at UGA in one of our seminars where we take a classic theorem. I think most recently, we did Maschke’s theorem, which is something about representation theory. And then you highlight every word in the theorem that you could change, and then kind of come up with conjectures based on changing some of those words, or questions based on changing some of those words. That's a really fun exercise in, kind of, mathematical remakes.

EL: That does sound fun. And I mean, I think that's one of the things that you learn, especially in grad school, is just how to start looking at statements of theorems and stuff and seeing where might there be some wiggle room here? Or where could I sub out a different space or a different set of assumptions about a function or something and get something new.

DL: Right, exactly. Yeah, definitely. With Dirichlet’s theorem, that happens so many times.

EL: Yeah, well, that's very fun. Thanks for bringing that one up. Thinking about it, I’m a little surprised that we haven't had it already on the podcast.

DL: Yeah, it's classic.

EL: Yeah, it really is.

KK: So we also like to give our guests a chance to plug anything that they're working on. You're very on Twitter.

DL: Yeah, that's right. You can you can follow me @littmath.

KK: Okay.

DL: So what do I want to plug? I think aside from Sherlock Holmes, who maybe needs no plugging, first of all, I would like to plug the Ava DuVernay documentary 13th, which I really liked and I think everyone should should watch.

EL: Yeah, and I saw that's free on YouTube right now. I don't know if that's temporarily, but I’m not a Netflix subscriber.

DL: Yeah, it is on Netflix. And yeah, I don't know if it'll be available on YouTube but for free by the time this comes out, but probably a nominal cost. In terms of things I've done that I think people who listen to this podcast might like, I did a Numberphile video about a year ago on the on it one of Hilbert’s problems about cutting up polyhedra and rearranging them that someone might someone who likes this podcast might enjoy. So if you google “Numberphile the Dehn invariant,” that’ll come up.

EL: Oh, great.

KK: Cool. All right.

EL: We’ll put links to those in the show notes. Yeah.

KK: All right. Well, thanks for joining us.

DL: Thank you guys so much for having me. This was a lot of fun.

KK: I learned something. I learn something every time, but I'm always surprised at what I'm going to learn. So this is this has been great. All right. Thanks, Daniel.

DL: All right. Thank you so much.

On this episode of My Favorite Theorem, we were happy to get to talk to Daniel Litt of the University of Georgia about Dirichlet's theorem on primes in arithmetic progressions. Here are some links you might find useful as you listen:

Litt's website
Litt's Twitter profile
More about the Dirichlet theorem from Wikipedia
Tom Apostol's number theory book
The article Evelyn wrote about surprising biases in the distributions of last digits of prime numbers
Michael Chabon's novella The Final Solution: A Story of Detection
Litt's Numberphile video about the Dehn invariant
Ava DuVernay's documentary 13th