So Brady Yitang Zhang, the famous Chinese
mathematician, well he's done it again. It's about 10 years ago we did a video on the twin
prime conjecture which was solved - well not solved entirely but sort of a big progress was made by
this unknown Chinese mathematician. It came out of literally nowhere and made huge progress. So just
to recap what that was about, so the twin prime conjecture, that's where you know we've known for
many many years since the days of Euclid that there are an infinite number of primes. What the
twin prime conjecture says is that there are an infinite number of primes that differ by just 2.
So if p is a prime then p + 2 would also be a prime. And there should be an infinite number of
those guys. Now Yitang didn't prove that result but what he did prove was that there are an infinite
number of primes that differ by 70 million. So that's still a, you know, a finite number. And
then since that remarkable proof, mathematicians have got together, they've collaborated in
something called the polymath project, and they've managed to bring that gap down to I think
246. So now we know that there are an infinite number of primes that differ by 246. This is all
down to Yitang Zhang; but now he's done it again in another big result, you know, possible big
result, in analytic number theory. And it has to do with something called Siegel zeros.
- (Brady: Seagull zeros?)
- Siegel zeros, or Landau-Seigell zeros. (What are they?)
To really understand what this is we have
to go back to the Riemann hypothesis; which I know you've done videos about before. So just to quickly remember what that's all about; so you know, you think about this- the the uh the so-called zeta
function which we've we've seen before in lots of varyingly controversial contexts. This is where you
take the sum over n greater than equal to 1 of n to the minus s. So this would be like, you know, if
you'd have 1 to the minus s, plus 2 to the minus s, plus 3 to the minus s and so on including
all the integers. So this is known as the Riemann-Zeta function. You know, for example, we know
that if we set this number s to -1 then we have the sum of all the integers, or all the
positive integers, and we know what that gives us: without any controversy it gives us minus 1/12.
- "The answer to this sum is remarkably minus 1/12." But this is this is a very interesting function
in analytic number theory; we know that for- if this s, if it's real part is bigger than 1 then this
this sum will converge. And we can extend the definition of this function to the whole- to all
complex numbers using this sort of mathematical process called analytic continuation. So this
thing can be defined in principle for all complex numbers. What's really interesting about this guy
is is where does it vanish? There's an interesting result that you can link this to prime numbers.
This is a famous result due to Euler that if I go through through all the prime numbers like this,
if I take products of prime numbers like this: to the -1, 1 minus 3 the minus s
to the -1, 1 minus 5 to the minus s. So 5, minus 1- and so. I take
all these products, so I take these combinations of prime numbers, so these are all prime number
entries sitting in here, and I can carry on all the way up to infinity - this is also related to the
to the zeta function. This is a result due to Euler. So I think you've discussed this before right?
That this- so knowing about this function tells you about prime numbers, and how prime numbers are
distributed. One of the interesting things to ask is where does this function vanish? We can draw the
complex plane, okay; so this is the real part of s, this is the imaginary part of s. If I draw the line
where s equals- this is the this is 1 here, 0 here, so this is the line where the real part of s
is 1, this is the line where the real part of s is 0. And we know the the function converges
in this region here to the right of 1. I can continue the results to the whole complex plane. I
know, it turns out that you can show quite easily, that there are 0s at -2, -4 and
basically all these sort of negative even values. This is where there's- we know- these are called
the trivial zeros of the zeta function. They're not so interesting. What's interesting are what's
called the non-trivial zeros. And it's known that those non-trivial zeros will lie in this
critical strip between 0 and 1. The famous Riemann hypothesis says not only do they lie in
this strip but they actually lie on one particular line - where the real part of this complex number s
is 1/2. So the claim of the Riemann hypothesis is that all the non-trivial zeros of this very
important function which is related to prime numbers will lie on this line here. So they'll
all be here, here - wherever they are. That's the claim of the Riemann hypothesis. So now one can
generalise this- have something- generalise this idea to something called the generalised Riemann
hypothesis, it's very similar. So what we do is we just generalise this function a little bit. So
what we do is instead of considering these sums over n to the minus s, we consider a related set
of sums. So it's still very similar, so you still have your n to the minus s's like we had before,
but now we weight it with this weighting factor. Basically it's very similar to what we have
before but they're just slightly weighted, each contribution is slightly weighted by this number chi.
So if you want to write it out in full it would be chi-d of 1(1 to the -s) plus chi-d of 2 (2 to
the -s) and so on, right. These are called the Dirichlet L-functions. And these guys, these
factors, well these are the sort of new ingredient that we have now. They have various properties; of
course these functions that sit out front here, they basically, you know, they they take integers
n to some complex number okay, and that's what they are. They're just some function that does that.
They have lots of properties that you can discuss but we don't want to go into all the details. The
most important one is what this D represents and that's called the modulus. And that's just the
statement that whatever this function is, it's periodic. So if I take some integer and then I
add D to it then it doesn't change the value of what this function does. So if I- for example
if I evaluate this function at n equals 1 then I get the same value at D + 1; you see
what I mean? So it's just- it's periodic, it just repeats okay? So examples of this thing, right? So if
the modulus is- if D equals 1, if the modulus is 1, then this turns out has to be the trivial-
what's called the trivial character, and that means that it's just 1 - it's always 1, 1
everywhere gives us back this. If the modulus is 2 then you- then the the only thing you can
have is is what's called the principal character; and what that is, that's either- that flips between
uh 0 and 1. And think it's um it's 1 for odd numbers and 0 for even numbers. And so it just
has that sort of weighting. If you go to higher D, higher modulus, you get just more complicated stuff.
It's just a mathematical construction. The details don't really matter. The point is this is kind of
like a generalisation of the zeta function okay? The claim is, of the generalised Riemann hypothesis,
is that for these guys, again the non-trivial zeros lie on this critical line. That's the generalised
Reimann hypothesis. So we've gone round about, now we have to get back to Siegel zeros. Siegel zeros
is a counter example to that. So Siegel zeros are zeros which don't lie on this line; in fact they're
zeros that potentially lie very close to this line where the real part is 1. In particular they'll
lie sort of around here; so they're real zeros which lie very close to 1 basically. That's the
idea of a Siegel zero. If a Siegel zero exists then you can prove all sorts of wonderful things.
So there's there's loads of sort of proofs you can do in analytic number theory that rely on the
existence of a Siegel zero. If it exists you can prove this; if it doesn't exist well you maybe
can't can't prove it. Here's an example: there's something called the Heath-Brown Theorem. It says
the following - one of the following has to be true: there are no Siegel zeros or the twin prime
conjecture is true. So one of those has to be true, okay? They can't both be false. So- so in
other words if you if you find a Siegel zero, if you can show that it does exist, then that means
the twin prime conjecture has to be true, you see? So so you can see that these the existence
or absence of these Siegel zeros is really important for proofs in analytic number theory.
Okay, so so this is what Yitang is thinking about. Okay now the the precise statement of where
the Seigel zero lies is the following: it should lie on the real line very close
to 1 - how close? Well that's part of the theorem. It should lie within a small distance here
which should go like c, which is just some number, over log D. So remember what these things are; c
is just a number, it's an absolute number, it it doesn't matter what it is it's just a number - it
doesn't depend on D. D is this sort of periodic property, the modulus of these of these characters
here. That's the claim of the Siegel zero; if it exists it should lie in this little tiny region
close to 1 okay? And that width of that is c over log D, right? If it exists, of course, the
generalised Riemann hypothesis is false as well. That's another thing right? Because you know if
that's the- if that zero is there then that's that's off the line, that's off the critical
line right, so it's another important result. What did Yitang prove? He didn't prove the
absence of these zeros, he didn't prove that. He proved- so what you would like to show is that
these zeros don't exist in any of this region here. (Eliminate that area.)
- You want to eliminate that
area, that's that's the that's the thing you're striving for. He didn't do that. He managed
to show that zeros didn't exist in a much thinner region, okay? And the region that
he was looking at, it had this width: Okay, so he was able to prove- so this is a much
thinner region, much closer to 1, and he was able to show that there were no zeros in something
of that width. Okay? Now you might wonder where the hell's that 2024 come from right? Well he-
he did this by proving a related result which included the number 2022 - which was of course this
year, right? So it's a bit of an arbitrary choice. He himself says he can bring this number down to
order a few hundred; whether he can bring it down to the 1 that you need to really capture this
statement about Siegel zeros and where they are and the thing that's then going to feed into
all these other proofs - well I talked to some number theories here at Nottingham and they were
saying well that's probably going to need some new input to do that. But it's still a big
breakthrough, it's still it's still sort of you know a real advance - if correct - compared to what's gone before.
- (He's narrowed the search field a bit?) Yes. So basically he's been able to show that
if you go really really close to 1 in this region here there are no Siegel zeros. But to
capture the full statement you need to prove it in this slightly larger region, okay? And as we
bring this number down that region is going to get bigger and bigger and bigger and bigger and
bigger until you capture the whole lot and then once you've got that then all these factors that
go into other proofs, and this that and the other, will really kick in. Even this result can have an
impact on some sort of other mathematical proofs. (Do you want there to be no Siegel zeros?)
- No I'd like there to be one. Definitely! Because it disproves the Reimann hypothesis right! Or the generalised treatment hypothesis.
- (That's bad news if we) (disprove the Reimann hypothesis)
- New stuff is fun, Brady! New
stuff you don't expect. So most- by the way most mathematicians would say that there's probably
no Siegel zero. It's generally- so the proof's- it's it's considered what's called uh illusory,
that's what- that's the terminology they use. So so you can prove this if if there is a Siegel
zero but it's an illusory proof because chances are there isn't a Siegel zero.
- (Isn't there a lot of
mathematics that will just crumble if we disprove) (the Reimann hypothesis? Like it'll be like, oh lots
of things we thought were true- Aren't there lots) (of proofs that sort of say 'this is assuming the
Riemann hypothesis is true'?)
- There are, yes, and but then this then goes deep into proof theory.
I mean you can have um statements which are which are based on the correctness of their Reimann
hypothesis and statements which are based on the sort of incorrectness of the Riemann hypothesis
actually. And then they just take into different branches of mathematics actually. And there's this
really interesting notion, you know, that people look at, which is like um trying to unpick the
theories of mathematics and trying to understand um what proofs rely on what assumptions and
therefore- and then if you break those what does- which branch of mathematics does that take you in?
So, you know, very loosely you might say well what if I say I only have real numbers? You
know what can I prove? Well then you can only prove so much, but then you can say, well I can
extend my axioms, I can extend the rules of the game to include complex numbers and prove
more stuff. And and then there's this whole game that you can play with with mathematics. So
you're absolutely right. It'd just be fun right if there is one! Oh there's my little Siegel zero!
How cool would that be?
- (They'dname it after you) (It could become the Padilla zero.)
- Well if I found
it, I don't think there's much chance of me finding it. It's in there somewhere right? But but you
know; so this of course has has implications in in physics as well, at least the distribution
of prime numbers. So so the zeros of the zeta function, you know, they've been connected to
um energy levels of heavy nuclei. So there's a connection there right? So understanding the
zeros of the zeta function can maybe tell you something about the energy levels of heavy nuclei,
they're sort of distributed, you know. It's really quite profound potentially. In terms of this this
result, this latest result that he's proven, um the jury is still out amongst the mathematical
world - as it should be. Maths has this brilliant uh sort of philosophy that what they do is is
it, you know, they really- the peer review process is incredible in mathematics. They take a very
long time to unpick each other's work and it's a really serious process. So that's going to take
at least a year probably. You know, people like Terry Tao for example have already commented
on the- on the on the manuscript saying that the um a certain sort of inconsistent equation-
equation referencing and sort of just things in the paper that are making it difficult to sort
of follow the proof and check it. So um, you know, it just needs a time to just to sort of tighten it up,
tidy it up, and then say, off you go guys now check it, see if you if you agree. And actually
it may be that they don't agree; I mean this has happened before right? It may be that they spot a
small error. But then what normally happens- it's such a healthy culture, what will happen then is,
rather than everyone going oh it's wrong they'll probably, you know, try to work together with the
with the person reviewing the paper and actually try and fix it and actually- You know this has
happened in the past with Fermat's last theorem and things like that of course. So yeah I guess
the jury is still out but it's potentially quite exciting. How do you build a brain that
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life. See all the details on the website. Well I don't know about Siegel zeros but the
search for Tony's book 'Fantastic Numbers and Where to Find Them' just got a whole lot easier because
it's been released in the US. You can see both the UK and US covers here on the screen. There'll
be a link to that in the video description.
