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.
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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.