Norman J. Wildberger
Let's crack the Riemann Hypothesis! | Sociology and Pure Mathematics | N J Wildberger
Unlisted
122K subscribers
<__slot-el>
<__slot-el>
Subscribed
3
Share
42 views Apr 27, 2024
Modern pure mathematics implicitly assumes that we are able to perform an unbounded, or infinite number of arithmetical operations in order to bring into being "real numbers" and "values of transcendental functions" such as "exp" and "log". We use these superhuman powers to reconsider J. Lagarias' equivalent reformulation of the famous Riemann Hypothesis concerned with the zeroes of the so-called "Riemann zeta function". This allows us to resolve this famous conjecture with a computation, which is essentially similar in difficulty to the calculation of cos(7) or zeta(5) or exp(H_6) where H_6 is the 6th Harmonic number. Fellow pure mathematicians: are we not able to think clearly about what we are actually able to do when it comes to mathematical computation? We don't need to pretend, just to obtain "pleasant results" that distort the real nature of the mathematical world. Let's rather replace philosophical justifications using obscure axiomatic frameworks with simple alignment with what our computers can do.
0:06
Hello, we're examining the curious sociology of pure mathematics where
0:11
strangely we believe really crazy things. Here in pure mathematics, these beliefs
0:17
are like foundational beliefs supported by very little evidence, but a lot of wishful
0:23
thinking and principle. Among these fantastic beliefs is the idea that we can go to infinity,
0:30
that we can take a limit, which ultimately amounts to we're able to do an infinite number of things
0:37
and complete the job and get an output that we can then use for further machinations. Today,
0:47
we're going to examine the consequences of this point of view by cracking the most famous
0:54
unsolved problem in mathematics: the Riemann hypothesis. So I'm going to tell you very roughly
1:00
what the Riemann hypothesis is, in a way that's hopefully accessible to a wide audience. Then I'm
1:07
going to talk about an alternate formulation of it, due to Jeffrey Lagarias, one of many alternate
1:15
reformulations. And I'm going to show you how we can actually crack this problem by actually
1:23
making a computation, that is assuming these superhuman powers that pure mathematicians ascribe
1:32
to themselves. All right, so we're going to have a little bit of fun today, so let's start with the
1:37
zeta function, which is the principal object in the Riemann hypothesis. So this is ostensibly a
1:44
function, and you start by taking a natural number like 2, 3, 4, etc., and forming this sum. You look
1:53
at the sum of reciprocal powers of the natural numbers, so sum from k=1 to infinity of 1/kn. Now
2:15
it's important to recognize this is not a finite sum. Okay, this is meant to be an infinite sum,
2:21
so we do not stop increasing the sizes of these denominators, they just keep on getting bigger
2:26
and bigger and bigger. So correspondingly, these terms that we're adding get smaller and smaller
2:32
and smaller, but we still have to keep on adding them. So there's an infinite amount of work to be
2:37
done in creating such a sum. And pretty well every number theorist in the world, every analyst in the
2:46
world, says "Yes, this makes sense. This is a bona fide definition. This really does define what's
2:52
called a real number, and we're in a position to talk about this real number," and that's giving us
2:59
the so-called zeta function. Now I have to object. I don't believe this. I don't buy this. For me,
3:07
arithmetic is what you do with a finite number of objects. You perform a finite number of
3:12
calculations, and then you get an answer. So for me, the idea of computation is the same as what a
3:20
computer would do, right? A computer also performs a finite number of computations, and then gives
3:26
you an answer. So me and the computer world are quite aligned. So I may not have a lot of
3:32
mathematical colleagues that agree with me, but I have a lot of computers that are very much on the
3:37
same page with me. Okay, so let's have a look at an example, right? Let's consider Zeta of five.
3:45
So that means we replace the n here with five. We're talking about 1 over 15 + 1 over 25 + 1
3:49
over 35, etc. Okay, so as a warmup, why don't we go a little ways down this infinite road, maybe to
3:59
15? So let's take the sum from k=1 to 15 of 1/k5. So we're going to take all the terms from 1 over
4:07
15 all the way up to 1 over 155. Well, that's a finite sum, that's something that we can do,
4:16
that's something our computers can do. In fact, my computer has done it. There is the
4:20
result of the first 15 terms of this series as a fraction, because that's what it is. It's a
4:26
bunch of fractions added together, so you get another fraction, but it's not so easy to tell
4:32
how big this thing is exactly. So my computer can also convert it to an approximate decimal. Okay,
4:41
so since it's a fraction, the decimal expansion would be repeating, presumably with quite a long
4:47
period. Okay, but um, at first it looks like this. So it's 1.0369234, etc., etc. Okay, so that's
4:59
up to 15. Now you can kind of see that you know, you don't have to go too much further,
5:05
and these denominators here are getting pretty big because we're taking fifth powers.
5:09
So, these numbers that we're adding here are really small, and they have big denominators.
5:13
So when we're doing this fraction arithmetic, this thing is really getting big. Okay,
5:18
so if we watch this thing on some, say, infinite screen, this thing would be getting bigger,
5:25
bigger, and bigger with every step as we add more and more terms. But there wouldn't be a
5:31
very clear sense of this thing converging to anything. We wouldn't be able to see, "Ah,
5:36
it's approaching something." It's only when we convert it to like a decimal like this that we
5:40
have a sense of this thing, you know, approaching something. Because as we add more and more terms,
5:46
what's going to happen is that the initial terms here are going to sort of stabilize, and we're
5:49
only going to be sort of changing the entries which are further and further out here. Okay,
5:54
so there's some sense that as you keep going, this decimal sort of converges or stabilizes towards
6:00
some infinite decimal. So that's the usual story. It's a story that I don't buy because in fact,
6:09
long before you get to infinity, long before you get to infinity, say if we replace this 15 with,
6:16
say, 10 to the 10 to the 10 to the 10, just a stack of four tens, one on top of the other in
6:24
exponential fashion, already at that stage one can easily see that the sizes of these fractions
6:34
that we're getting are already far beyond the size of the known universe to encompass, even
6:44
if the entire universe was used as a massive hard drive, with data points crammed in on the Planck
6:54
scale. So there's no question that long before we get to infinity, the thing that we're calculating
7:02
here blows up, and you know, the universe that we are in is no longer capable, even theoretically,
7:10
of encompassing that. So the pure mathematician who wants to justify this has to have some shift
7:17
in their minds. They have to say something, "Well, okay, what we're going to do is we're going to
7:21
kind of pretend that this is all happening in some alternate universe. We're imagining some other
7:27
universe, say, outside our universe, which has the very convenient property that it's infinitely
7:33
large, not like our universe. Our universe is finite, probably." Okay, no, we have to imagine
7:39
these universes which are infinitely large, and therefore they contain enough memory space to
7:45
be able to hold these huge things as they blow out of our universe. So there's a lot of, essentially,
7:53
religious, um, sort of dogma associated to this, right? It's almost like biblical proportions,
7:59
the kind of, um, you know, wonderful things that we have to postulate in order for this story to
8:07
make even a shred of sense. And it's far removed from computational reality, right? There's no way
8:14
a computer can deal with anything like this. Once this number here gets to the order of 10
8:20
to the 10 to the 10 to the 10, least of all when it actually is approaching infinity in a serious
8:25
fashion. But in a spirit of generosity, let us, for this lecture, embrace the standard orthodoxy
8:34
and pretend, as my fellow mathematicians do, that actually these things do make sense, and that this
8:40
series defining the zeta function is actually well defined. How does the story unfold after that?
8:47
Okay, well, the idea is here is a number line. There's zero, there's one, there's two,
8:52
there's three, and we think of this zeta function as giving us values for any point in here. Right,
9:01
well, we started with five, and we got something like 1.036927. Now we're assuming,
9:07
yes, there's an infinite decimal here that's associated with the zeta function's value at five,
9:13
and correspondingly, you know, we could get numbers four, three, and two, etc., and rather
9:18
remarkably, this connects with beautiful formulas of Euler, which I also have some qualms with,
9:25
but I do admit that there's something really beautiful happening under the bonnet here, but we
9:29
just haven't been able to say yet what it actually is. But his results amount to the fact that you
9:36
can create formulas for the zeta function's values at these even integers in terms of powers of Pi,
9:43
like this one's Pi squared over six, this one's Pi to the fourth over nine, and some other remarkable
9:50
numbers coming from Euler. But the odd ones here are much harder to say anything precise about. But
9:58
we're going to assume now that this zeta function expression works not just for natural numbers n,
10:06
but even for fractional values. Okay, so those one over integer to the n, we're now going to suppose
10:16
that even if we replace the n with a fraction like maybe 2 over 9, that we can still write out that
10:24
series and make that calculation. So that's a huge additional assumption, because now we're supposing
10:32
that if we have something like 4 to the 2 over 9, that we're getting something irrational here,
10:40
we're getting something that's not itself a finite operation. Right, to calculate 4 to the 2 to the
10:46
9, it's like the ninth root of 16, that itself is an infinite operation, can't be done in a finite
10:54
amount of time, your computer cannot do that exactly. So the extension of the zeta function
10:59
to say rational values here on the number line is already a very big extension, which we're also
11:06
going to happily assume we can do. And then it turns out that if we think about this number line
11:11
as being embedded in the complex plane, so we have another direction with this complex number i whose
11:18
main property is that i squared equals -1, then it turns out that this zeta function ostensibly
11:23
can sort of be extended to complex values. So we can associate a meaning to this infinite series
11:30
even when the exponents n that are appearing in the denominators are replaced with more general
11:37
complex numbers. Okay, and it turns out those series converge in a sense that the analysts like
11:46
sort of in this portion of the complex plane. Okay, then it gets a little bit complicated.
11:52
There's something called analytic continuation that allows you to take certain formulas for this
11:57
thing over here and sort of extend that formula to extend the function's values essentially to
12:03
the entire plane. Okay, so turns out to be then some kind of symmetry already observed by Euler
12:09
I guess, maybe Riemann, about this critical line where the horizontal component is a half, that
12:18
plays a very important role in this subject. So now we're imagining this complex-valued function
12:24
which is an extension of the zeta function that we first were considering to complex number values,
12:31
and it turns out that it has some interesting things. So at this point one, there's a kind of
12:35
a blow-up at the negative even numbers like 2, -4, etc. It turns out that there's zeros. Okay,
12:45
this analytic continuation has some zeros there. They're called trivial zeros because they're quite
12:51
predictable. So the Riemann hypothesis then amounts to the following statement:
12:57
that all the other zeros, like that is, the ones that are not these trivial ones on the negative
13:03
axis here, they are all happening on the line where the real part is equal to a half. Okay,
13:11
so you know, there's a whole bunch of these zeros ostensibly, and they're all on this line here.
13:15
Now, this is not the scale. They're not exactly like this. I'm just picturing them to give you an
13:19
idea. Right, that's the main um conjecture or the most famous conjecture or unsolved problem in mod
13:29
mathematics: is that this extended Riemann zeta function has all its non-trivial zeros on this
13:36
critical line. And when you state it this way, it sounds really obscure and so on. But in fact, um,
13:45
this has huge uh connections with prime numbers and all kinds of interesting questions in number
13:50
theory. And I should say, you know, that there's, I don't want to say that the Riemann hypothesis
13:59
is nonsense. No, it's really uh, tackling um, a really delicate and um,
14:07
remarkable aspect of the world in which we live. Okay, I have no question about that.
14:13
But I question the language and the uh, sort of overarching analytic framework in which it
14:22
is currently presented. Right, so I believe there's a very interesting phenomena there,
14:26
but that we have not yet put our f on how to even express this phenomena in the right way. Okay,
14:34
so I just want to make it that clear that there is something going on, there's something very deep,
14:39
very profound, very interesting going on with the Riemann hypothesis situation. It's
14:45
just that we haven't yet understood how to even formulate it in an exact precise way.
14:52
Now, because this problem has attracted so much attention since it was formulated
14:56
by Bernard Riemann in the middle of the 19th Century, many people have discovered alternate
15:02
formulations of it, which again is some kind of indicator of the intrinsic interest of this
15:08
problem. It is really a very interesting problem, whatever that actually is. Okay,
15:14
and here is one of actually many reformulations of it. It's one that I particularly like.
15:19
I think it's very attractive. It was found by Jeffrey Lagarias in 2002 in the American
15:25
Mathematical Monthly. It's written up and it's um, concerning these numbers called harmonic numbers,
15:30
which are just rational numbers. Okay, so they are sums of reciprocals, just 1 over j,
15:36
j equals 1 to n. It's like 1 over 1 + 1 over 2 + 1 over 3, that's the third harmonic number Hn. Okay,
15:46
and here is his reformulation of the Riemann hypothesis: that for any natural number n (1,
15:53
2, 3, etc.), this inequality holds. On the left-hand side, we have Sigma of n. It's a
16:00
number theoretical function; it's the sum of all the divisors of n. Because if you have a number
16:06
like six, its divisors are 1, 2, 3, and six. Those are all the natural numbers which divide six, and
16:16
the sum of those things is 1 + 2 + 3 + 6, that's 12. It's actually quite interesting because that's
16:22
exactly twice the number that makes six a perfect number. Okay, so that's what's on the left-hand
16:29
side, that finite sum. It's just a natural number. Okay, on the right-hand side, we have this Hn.
16:36
Okay, n is the same number there, so it's the nth harmonic number. And then this expression here
16:41
involving the product of the exponentials of Hn with the logarithm of Hn. So I have to remind you,
16:49
you know, what are exp and log? Well, these are also functions, actually very familiar sort of
16:55
elementary functions in analysis, but still they have this infinite nature which I um,
17:03
object to ultimately. But anyway, here is the standard definition. So exp of x is
17:08
defined to be 1 + x + x2 / 2 factorial + x3 / 3 factorial. 3 factorial means 1 * 2 * 3. Okay,
17:15
and then you keep on going, and again, this is an infinite series, so it keeps on going. So
17:20
you have to take all of these terms and you have to perform this infinite amount of arithmetic.
17:25
Okay, the terms are typically getting smaller because these factorials get bigger,
17:29
so the terms are getting smaller and hopefully you're converging to something.
17:34
So that's the standard definition of the exponential function. The log function,
17:39
well, here's one formula for it: log of (1 - x) is also given by a series, something like this:
17:44
x + x2 / 2 + x3 / 3 + x4 / 4. There's no factorials, just simple denominators there.
17:52
And this particular formula is known to sort of work only when x is in a nice range close to zero,
17:59
when it's between -1 and 1. So you can think about this as being the definition of log,
18:05
and other values of logs can be sort of relatively easily converted to this one.
18:11
Okay, so let us temporarily, for the purposes of this talk, let me assume these superhuman
18:18
powers that analysts and number theorists like to give themselves, which is the ability to actually
18:26
perform calculations like this. So I'm going to suppose that, yes, I actually can make sense of
18:31
this, that this is a well-defined thing for me. I want to find exp of something, I just
18:37
take this series, I plug in values for x here, and I perform this infinite amount of work. Yes,
18:45
it requires going outside the universe, because again the same story happens as these numbers go
18:50
further and further: everything gets big, things blow up in our universe, there's no way we can do
18:54
this in our own universe. So we have to imagine this all happening in some alternate universe
18:58
which has an infinite amount of room in it, infinite amount of computing power available,
19:05
infinite amount of patience, and time. Well, maybe not time. We're going to assume that we can do all
19:10
this in a finite amount of time. If you or I had to do this, well, obviously we would just have
19:17
to keep on going, but we're going to assume that no, we can do this in a finite amount of time. And
19:21
same thing for this expression here. So that's really the core, that assumption is in fact so
19:28
powerful, okay, that I'm trying to convince you it actually essentially trivializes so much of
19:34
mathematics, and it essentially trivializes this problem too. Okay, so this assumption,
19:40
which is built in to even understanding and appreciating what this statement is,
19:44
this assumption is already enough to essentially kill these problems like this.
19:51
I also should have said that another part of the thing is that there should be equality,
19:56
strict equality when we have equal sign there, um, precisely when n equals 1. When n is one,
20:01
this is one, this is one, this is something and this is zero, so we get 1 equals 1. Okay,
20:08
but otherwise there's strict inequality here, so this is Lagarias' restatement, and let's see how
20:16
we can address this because this is a lot easier, right? We don't have the complex numbers, we don't
20:21
have this mysterious Z function, we don't have to perform analytic continuations, we don't have
20:25
to know fancy formulas about how these functions sort of behave with respect to the critical line.
20:32
It's a relatively simple expression, assuming we are allowed to have infinite computational power
20:40
to get at this exp and to get at this log. Those are the two big assumptions that we're
20:46
actually making here. All right, so please watch carefully, nothing up the sleeves. All right,
20:50
so here is what we want to establish: that for all natural numbers n bigger than one,
20:56
that we have this strict inequality between this sum of divisors and this expression Hn plus x
21:02
Hn times log Hn. Let's refer to the statement's truth value as P of n. So P of n will be a Boolean
21:11
expression, it's just either zero or one depending on whether this statement is false or true. So for
21:19
a given value n, if the statement is false then P of n has value zero, and if it's true it has
21:25
value one. So to make things explicit, let's have a look when n equals 6. When n equals 6,
21:31
well, we've seen that this thing here, the sum of the divisors, that's equal to 12. Hn is then
21:36
the sum of 1 + 1 over 2 + 1 over 3 + 1 over 4 + 1 over 5 + 1 over 6, that is 49 over 20.
21:44
So what we have to prove is that 12 is less than 49/20 + X of 49 over 20 * log of 49 over 20. So,
21:56
for this to make sense, we have to be able to compute X of 49 over 20,
22:01
right? That's implicit here, that this actually has a legitimate meaning, not in an approximate
22:07
fashion like an applied mathematician would want. Okay, an applied mathematician does not care
22:12
about infinite decimals; an applied mathematician thinks about this differently. It's just a finite
22:17
number of terms, a convenient truncation of that infinite series that I showed you. And similarly
22:23
for the log. No, but we're approaching this as pure mathematicians, so we're going to suppose,
22:28
because of these superhuman powers that we've currently adopted, that we know exactly
22:32
what this means and that this has a legitimate meaning for us. Yes, that series can be totaled,
22:40
that infinite computation can be done, and there's the result, and there's the result.
22:47
Okay, so we get some infinite decimal and by actually taking truncations, okay,
22:55
my computer can't do this calculation but it can do it approximately. It can take truncations of
23:00
this and truncations of that, reducing us from infinite arithmetic to finite rational arithmetic,
23:07
and then it can give this approximation to the right-hand side: 12.834178, etc.,
23:15
and that's enough for us to see that yes indeed, this 12 is less than this value
23:22
here. Now, if I was really able to do this infinite amount of work I could actually
23:29
compute this exactly. I could compute this exactly and I could obtain this thing here
23:33
not just to eight 20 decimal places but to an infinite amount of decimal places. That's
23:39
what I'm assuming just temporarily that I can do that. Okay, so conclusion is that when n is 6,
23:46
this Boolean expression P of 6 has the value one because the equality of Lagarias is true.
23:54
And now, ladies and gentlemen, taking the powers that I have been temporarily given, I am going to
24:02
compute this crucial quantity. Let's call it W sub RH, RH for Riemann Hypothesis. It is the product,
24:12
here's another sign, like a summation sign, it means take a product of things from 2 to infinity
24:20
of P of n. So, what does that mean? Well, it means that I'm able to do an infinite number of things,
24:31
right? So, I've just calculated P of 6 for you but I'm going to start now with P of 2. I'm
24:38
going to calculate P of 2 making an infinite computation, I'm going to calculate P of 3,
24:43
make an infinite computation, check the equality P of 4, P of 5, P of 6, there's an infinite number
24:49
of tasks that I have to do, each of them requires an infinite amount of work, but hey, I can do an
24:54
infinite number of things in a finite amount of time so I can do all of those things. So,
24:58
I can make a list, I can tabulate what is P of 2, what is P of 3, what is P of 4,
25:03
and I have this infinite list of these Boolean truth values, right?
25:07
There, each of them is either one or zero. And what I'm doing here in this final step is I'm
25:16
just multiplying all of these together. Another infinite task, but hey, we're all capable of doing
25:22
infinite tests, aren't we? Sure, so 1 * 1 * 1 * 1. Well, if they're all ones, we're going to get one.
25:30
If one of them is zero, the whole thing is killed and we get zero. So in fact, once we get to zero,
25:37
we know what the output is. Zero. We don't even have to carry on to the end, but carrying on to
25:42
the end doesn't bother us. It's just doing an infinite number of things. Hey, we're assuming
25:47
that, in even defining this, in even accepting that this is a valid object. Okay, so this is
25:53
a number, this is the result of a computation, okay, which in principle is really not that much
25:58
different from this quantity here, X of 49 over 20, whatever that is. That is a legitimate object
26:07
in modern number theory and analysis. If a problem ends up with the answer X of 49 over 20, everyone
26:16
will say, "Oh, good, we've got an answer, you may stop." That's considered a final complete answer.
26:25
Nobody asks, "Yeah, but what actually is this?" Show us the infinite decimal? No,
26:30
everybody is polite enough to know not to ask that question. Okay, except perhaps for me,
26:36
but everybody else is accepting that we're just applying exactly the same criterion here. There's
26:40
no need of asking what is this actual value. I don't have to tell you, this is an expression
26:47
for the truth value of the Riemann hypothesis. Is it true or false? The answer is, it is this.
27:01
So the consequences here, I hope that what you're going to get out of this is that this assumption
27:07
that we can do an infinite number of things, which is implicit in not just the definition of X,
27:14
the definition of log, or the definition of the zeta function, but it's implicit in so
27:19
much of modern mathematics, not just analysis and number theory. This assumption is so strong that
27:26
it basically trivializes a lot of mathematics, and somehow people haven't put this together. People
27:33
have not wanted to see that this particular assumption that we can do this has all these,
27:39
you know, really distorting consequences. But that's just the logical reality of it. So,
27:45
if we want proper mathematics, we have to become much more modest about what we actually
27:52
want to assume about ourselves, and a good way of proceeding is to, well, why don't we just assume,
27:58
assume that we can do what we actually can do? And if there's any question about that, why don't we
28:05
just peg ourselves to what our computers can do? That's very cut and dried, and we can all agree.
English
AllFor youRecently uploadedWatched