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0:00
So we're going to be talking about uh some new world records uh going 23% beyond the Riemann Hypothesis. So the Riemann Hypothesis is one of the great problems in mathematics with a million dollar prize if you solve it. But before I get ahead of myself, let's just talk about our favorite numbers - the primes. So the sequence begins just with 2, 3, 5, 7, 11, 13, 17, 19, 23 and so on. So we know by Euclid the sequence of primes goes on forever but famously the primes
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are really known for being mysterious and having a lot of complexity so maybe a simpler thing we could look at is just the last digit. So consider just lopping off the other digits and just considering the ones at the end. So it starts off the same; for the single digit primes 2, 3, 5, 7 and once we get to 11 we just keep the 1 at the end. 13 we just keep the 3 at the end, 17 we just keep the 7 at
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the end and so on. So Brady I'm not so fast writing all these numbers down, especially if they go on forever, maybe with some help of a computer maybe we can speed up this process. Let's just consider looking at the list of prime numbers up to 100. So it starts off as before - 2, 3, 5, 7 - but then it goes up to 97, so that's the largest prime up to 100. So we look at the last digit of these primes, it starts
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off again 2, 3, 5, 7 - but then at the end when we're throwing off we get 89 becomes a 9, 97 becomes a 7. The advantage of a computer here is that we can just change one character and instead of looking at primes up to 100 we could look up primes to 500 and just generates that list uh pretty readily. And if we do the same thing, just looking at the last digit, we get this uh longer table of
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data. As a mathematician once you see a table of data your ears perk up and you ask uh you know are there any patterns in this data? Um so Brady maybe if we stare at this table maybe some thoughts are going to emerge; which numbers occur frequently? Are there any numbers that don't occur? You see anything? - (Brady: There are no even numbers except for the 2 at the start.) - And that's because?
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(It would be divisible by 2.) - Right right. So if a number ends in a 2, ends in an even number, then the whole number itself is even. By a similar argument, maybe a bit more subtle, there are no 5s in this list after the the first 5 because if a number ends in a 5 then the whole number itself has to be divisible by 5. - (Same for 0?) - Yeah so 0 is a double whammy; it hits us both with being even
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and also being a multiple of 5. So what we've just shown here by this argument is that we only have a few options here for the digits that can appear in this list. So so after the single digit numbers eventually it's either going to be a 1 or a 3 or a 7 or a 9. And here the point that you've just figured out is that the number 10, the base is equal to 2 times 5 so we've just eliminated
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all the digits that have a 2 in it or a 5 in it. - (Okay so we've got these four numbers; the next) (question I guess people ask is which are most common?) - Yeah I mean that's that's the that's the question. So a natural thing to do is to maybe plot these numbers on a histogram; so let's look at the numbers up to 100 and look at the digits. So as we said before there's going to be only one instance
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of a 2 in this, and we put them into bins, say one instance of a 2. Similarly there's only one instance of a 5 and all the other numbers up to 100 are either going to have a 1, a 3, a 7 or a 9. One of the great advantages of a computer is you can just change one character again and generate a lot more data. So if we look at numbers up to 1,000 we can get these bar charts where these single entries are getting slimmer and slimmer and these other four bars are getting taller and
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taller. And maybe if we go very far, let's say we go out to a million, we look at the primes up to a million and look at their last digit there's just a tiny sliver corresponding to 2; a tiny sliver corresponding to 5 and then all these other four options seems to be almost the same height. So roughly uh a 25% chance of getting a 1 as a last digit of a prime, same for 3, same for 7, same for 9. So there's an equal chance to get any of these four options. That is no
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coincidence Brady, this is the prime number theorem in action. We've been playing this game, this fun little game, uh looking at the last digit, but really we've been using the fact that we have you know 10 fingers on our hands; but we can look at other bases. You know, for example computers work in base 2, all their uh data stored with 1s and 0s, so we we can play the same game uh looking
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at the last digit uh in other bases. Let's just change one line of code and now uh maybe we get something uh a bit less interesting here so we get uh our first uh entry is a 0 corresponding to the prime 2 and then all the other primes have their last digit 1 in binary. So this just means that all the primes after 2 are odd. Maybe this wasn't as interesting but the next interesting thing we could look at is maybe uh base 3 so we just increase by one and we look at the prime
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written in ternary and just extract the last digit. So as before we just have a 0 corresponding in this case to the number 3 itself but all the other options for the primes are going to be either a 1 or a 2 for their last digit in ternary. And so this really again suggests, why don't we play the same game, now look in base 3, maybe the numbers up to 1,000 - there's going to be one number, namely 3, that has the last digit 0 but all other numbers are either going to have a 1
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or a 2. And again these are very much dominating. Maybe if we go up to a million again we see that the heights of these two options - almost 50/50 split. And again this is no coincidence Brady. So you could pick your favorite base, I don't know say 101, uh and you can have play a similar game: you look at all the last digits of primes uh in base 101 let's say numbers up to 5 million and you
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can get a nice uh roughly uniform distribution. And maybe there's a bit of wiggle room at the top but as the numbers get larger and larger the wiggles start to to to damp out and you get this result. So Brady this is uh kind of a a fun illustration of one of the deep results in mathematics, the prime number theorem. So people on the channel may be familiar, in this case the prime number theorem for
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arithmetic progressions; which says the last digit of a prime in any base you pick eventually tends to have an even chance. And that's of course unless there is some clear reason why there's a violation. So in the case of an even digit; so as long as the digit has no shared factors with the base we expect an even spread across all options. So maybe to be a bit more precise as a mathematician
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I just want to introduce some math notation: pi of x here is the number of primes up to x and the more precise count that we are interested in is looking at the number of primes up to x that have last digit a in base b. - (Pi has nothing to do with pi pi?) - Right. So this is one of those things where mathematicians uh are overloaded with notation at any time and maybe at some point in 1700s
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people decided you know this was a a convenient letter to to use for primes; the same uh the same consonant uh, but yeah it's kind of a stretch. So this is the number primes up to x that have last digit equal to a and this is written in base b. The result of the prime number theorem tells us in a more precise way that each of these options for digits occur with roughly equal frequency. So
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for any choice of digit and base, pi of x b comma a we have this approximate relation that this count is asymptotic to pi of x over phi of b. So here phi of b is the Euler phi function which just counts; so phi of b is just counting the number of possible digits we can use. So the number of digits, a, that have no uh shared factors with b. So in maybe the mathematical lingo we say that the gcd, the
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greatest common divisor, of a and b is equal to 1. Yeah so this is a a nice piece of mathematics that holds eventually so as x gets larger. But a natural question Brady is maybe you're a little bit impatient and you want to know, you know, how long do I have to wait uh for this to kick in and those bars start to even out? - (I mean that depends how you define even out as well doesn't it?)
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Right I mean so so there are certainly very technical- like various levels of technicality how you want to be precise; but morally speaking all we want to know is how long do we have to go along the number line to really start to see uh these numbers evening out. But we've now just with this uh seemingly innocuous question, really gotten thrown in the deep end. Think of you know people playing around
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on the shore at the beach, maybe you go snorkeling and around some coral, but somehow you took a deep dive and somehow you're now at the bottom of the Marianas trench, you're at the deepest of the deep in mathematics - we've now gotten to the Riemann Hypothesis. Okay so I really want to narrow in on this this key relationship in the prime number theorem which is saying the digits really even out
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in their distribution among the primes, so let's give this a start here. So the Riemann Hypothesis really has to do with answering the how soon question about star. So the Riemann Hypothesis tells us that star will hold once we get roughly speaking uh x being larger than the base squared. So this is a key threshold; so in other words the Riemann hypothesis - or more precisely the generalized
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Riemann hypothesis - says that this relationship star holds for bases b that are less than the square root of x, as x tends to infinity. In other words the prime number theorem is telling us a a nice relationship that exists among primes for any base b we we choose; maybe 10, maybe 3, maybe 101. But the Riemann hypothesis is really telling us that we can get the same phenomenon occurring even if we allow the
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size of our base to grow moderately with x, with the size of the numbers. - (I've made videos about) (the Riemann hypothesis before - and I don't pretend to understand it - but I've seen graphs and critical) (strips and and all these different numbers but I've never heard anything about the last digits of) (numbers or bases) - Right. I mean there are many ways to approach this uh great trench of the Riemann hypothesis.
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So one of the kind of links with uh the other for- perhaps more standard technical formulation of the Riemann hypothesis has to do uh with the zeros of the Riemann-Zeta function so-called and their placement on so-called the critical line of 1/2. And it is no coincidence that the critical line of 1/2 is the same constant of 1/2 occurring in this threshold of the square root of x. So these are- these
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two relationships are intimately linked in this way. But I- I think this is a I guess a more down to Earth way to kind of talk about the Riemann hypothesis. - (Okay) - So obviously the Riemann hypothesis is one of these kind of white whales of mathematics that is still unsolved today. But amazingly in the 1960s mathematicians were able to prove an average version of this result.
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Bombieri and Vinogradov in 1965 they were able to show that this relationship star holds for almost all bases b. I'm meaning that this relationship holds for over 99.9% of b's up to this Riemann hypothesis range - square root of x. When I first heard about this result in uh my first year of uh grad school I thought it was- I mean absolutely stunning because uh I I somehow felt like the Riemann hypothesis was
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you know somehow unobtainable and no one's really been able to say anything about it; and so I feel when I first learned about it I was almost feeling like uh someone's keeping this from me. So I I really feel someone should have told me about this earlier. And the way I like to think about this result, this over 99% result, maybe an analogy with with cleaning cleaning tools. You know if you
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want to clean around the house, maybe you want to sanitize your hands, you get Purell. Or if you uh want wanted to clean your countertop you get Lysol. It says on the label if you read it, cleans over 99% of germs right. So this is their real real selling point; and this is because you know for work around the house 99% of germs is really more than okay. And it's only really in kind of extreme
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ideal circumstances like being a hospital, in a hospital where you need a a full sterile room, uh to get that 100%. And so by the same token as mathematicians uh the Riemann hypothesis is really this kind of Ideal scenario uh getting a full result that we want to hold but for most practical purposes as a mathematician, in my research, the Bombieri-Vinogradov theorem is this completely adequate substitute uh for application- for a lot of applications. - (I'm not having that Jared.)
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(I'm not having that. Because in the infinite- because in the infinite world of numbers 99% isn't) (that much. Like there's still a lot of numbers that it doesn't work for.) - That is definitely true and obviously this is part of the conceptual reason why we want to have a complete understanding; but in one of the kind of facets of the Riemann hypothesis is that people have proven many
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results uh with the initial caveat 'assume the Riemann hypothesis then..' And there were many results in the 1960s that went from a conditional uh hypothesis to uh complete ironclad theorem once this uh result came out; and it turns out in some technical settings it is enough to just know this 99% version to get the desired end goal if you were using the Riemann hypothesis implicitly in your work.
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(But if they found uh a power, an exponent, where Fermat's Last Theorem broke; there was like one number out) (there where Andrew Wiles- this building wouldn't be named after Andrew Wiles) - Yeah yeah. So you know as a mathematician you really want to get that complete understanding and exceptional cases can really uh be important, and so this also ties in with uh previous uh things on the channel about Siegel
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zeros which are these potential real uh monsters is uh trying to thwart the Riemann hypothesis. Before this I mean I just want to emphasize how kind of amazing I find this, because before this result no one really knew anything. Like you couldn't- uh you couldn't really say uh much at all, uh and really taking the statistical perspective uh has really been leading to a flourishing in in mathematical
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research. So this is well not again the the ideal scenario but for most practical purposes as a working mathematician around the house uh this this is really kind of an important uh tool in the toolkit. - (Okay I believe you) - So now we get- now we get to the real uh the real beans. Once presented with this result, and if we're if we're only looking at some of the bases and not all, a question comes up,
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another question. We asked how soon? But if we're only looking at some of the bases can we go even sooner? Can we get a rate of convergence faster than what the Riemann hypothesis is predicting? So can we get uh bases b that are beyond this square root x barrier? So in other words Brady I'm asking to go beyond the Riemann hypothesis. It was uh remarkable results in the 1980s that actually
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first uh made a proof of concept showing we have this- - (What is x again? Remind me what x actually is?) So x is the size of the numbers- the size of the primes were going up to. So sometimes it was like 100, sometimes 1,000 or a million. And this gets these- these uh bar charts get more and more refined as as x gets larger. But really the the meat and potatoes of the question is how soon?
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Can we go even sooner? - (That's how soon they start evening? But how are you defining evening?) Yeah so I mean uh you can measure - and for example look at the difference between uh the desired quantity and then the the expected average. And then- so you know in statistics you have the concept of a a mean or kind of the average of a distribution and then also the variance or the standard deviation.
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And so you can ask more technical questions about looking at the standard deviation, this is really what the Riemann hypothesis is all about. So this is kind of the second order question about the distribution of primes. So we have this of course uh result from 1965 which gets this uh Riemann hypothesis on average bound. But this was improved remarkably in the 1980s, getting exponents up to x
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to the 0.571. So the way to interpret this Brady, this breakthrough, is we went 15% beyond the Riemann hypothesis. And this result really for almost 40 years was the state-of-the-art going beyond the Riemann hypothesis until about two years ago where James Maynard uh actually extended this result. So in 2021 he found that there were some bases that you could take all the way up to x the 0.60. So
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in other words we can go up to 20% beyond the Riemann hypothesis. So after almost a four decade uh hiatus uh there's been progress on the problem by James Mayard who- fans of the channel. - (He's up-) (He's up there somewhere! His office is up there. We should have got him in here) - Yeah and so today I'm happy to say that we can get even further up to b up to 0.617 - in other words we can go 23% beyond
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the Riemann hypothesis. - (What does that mean?) - So this means that there are some bases b of up to this size going beyond the Riemann hypothesis range, for which the primes in that base have their digits being kind of nicely uh distributed as we would expect. So we only have to wait a shorter amount of time, 23% uh less time than we would expect from the Riemann hypothesis- - (Before they start flattening out)
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Before they- so so this is kind of a savings of time uh in one sense; in other words we can take more bases and this result is holding for for more uh base b expansions. - (Do we know what any of these bases are?) Uh so that's one of uh maybe the kind of aspects of this more statistical approach is that we're looking at families of numbers. And so one of the insights, all all the way going back to the
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Bombieri-Vinogradov theorem, is that if you look at collections of numbers or families of numbers together you can gain additional insights that might not be as apparent from looking at an individual number. So we can't necessarily pinpoint an exact b that this holds but we can still have a family uh that's of a larger size. And again maybe in in an ideal scenario we can try to prove the Riemann
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hypothesis and we can prove these nice results for every single b; but in many practical applications these kinds of results are are very useful. We'd actually expect this to hold for all the bases but that's again beyond what- so even if you prove the Riemann hypothesis this would be a question yet beyond what we can do there. So that's maybe far off in the distance for future generations to
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think about. But at least from a statistical point of view this is uh I feel making good progress. (We know that it holds for base 2, 3, 4) - Yeah so for any- for any individual fixed number, already the prime number theorem is telling us that we get this nice convergence. But it's only when the bases are allowed to grow moderately um we we want to be able to quantify that rate of growth. So uh already
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we know even by the Siegel–Walfisz theorem which was proven over about 100 years ago we can allow the base to grow logarithmically or very very slowly with x and not run into trouble, but really you you have to assume these very heavyweight conjectures like the Riemann hypothesis if you want an individual b to be growing uh like up to the square root of x. These kinds of results uh I feel
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have a very uh beautiful aspect where they connect so that the proof ideas that go into them connect a wide range of ideas across maybe seemingly disparate areas of math. So on the one hand uh some ideas that go into this relate to modular forms, these kind of exotic objects uh that actually ended up playing a crucial role for example in Fermat's last theorem proven by Andrew Wiles and has led
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to the Langlands programs. Um on the other hand uh it uses results from algebra geometry; uh more specifically the the Weil bound and the Deligne bound, from the so-called Riemann hypothesis for curves over finite fields uh which is itself another story for another time. But I just want to convey the sense that a lot of- uh there's a confluence of a lot of ideas coming together just to uh make
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progress on understanding the distribution of just the primes. And as a just a last shout out, so I wanted to say that there seems to be a kind of nice revival in this field of studying these questions uh in the last maybe decade where there before had been something something of a lull; I just wanted to shout out some people, who are who have been uh doing some interesting work and we've been having a nice discussion. So it's really
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good to see this uh flourishing of this this subject. ...100. Between 1 and 1,000, between 1 and 1 million. For any uh n between one and n using his zeta function. Which is absolutely astonishing because if you think about zeta function, something has to do with complex numbers and with analytic continuation and so on and so forth. So it's a particular branch of mathematics which is called
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complex analysis. But prime numbers live in a different branch of mathematics. - It tells us things we already suspect because we sort of believe the Riemann hypothesis is true but there are things we can't prove. So for example how many primes are there up to a trillion? We have a formula for that that we believe is true but we can only prove that formula if we know the Riemann hypothesis is true.
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