Welcome to the first Mathologer video of the year. Today it's about something very serious and so I'm wearing a totally
black t-shirt. You all like Numberphile right? Me too, except for this one video
over there in which they prove the infamous identity 1+2+3+...=-1/12 using some simple algebra that
even kids in primary school should be able to follow. Since this video was
published in 2014 over six million people have watched it and more than
65,000 have liked it. Unfortunately, pretty much every single statement made
in this video is wrong. And by wrong I mean wrong in capital letters. In
particular, as anybody who knows any mathematics will confirm 1+2+3+... sums to exactly what common sense suggests it should namely plus infinity.
And this video was not published on the 1st of April. Also, as we all know, the
Numberphile videos are presented by smart guys, in this case university
physics professors, who do know their maths and who are definitely not out to
mislead us. So how did they get it so horribly wrong and what did they really
want to say. Well, they started out with some genuinely deep an amazing
connection between 1 + 2 + 3 etc and the number -1/12 but in the effort to
explain this connection in really, really simple terms they just went overboard
and ended up with an explanation that is not just really simple but also really
wrong. Well 6 million views later and the comment sections of all maths YouTubers
are being inundated by confused one plus two plus three comments that are a
direct consequence of this video. For mathematical public relations it's a
disaster. (Marty) It's THE disaster. (Mathologer) Yeah, it's THE disaster. And so I think it's a good idea to have another really
close look at the Numberphile calculation step by step, state clearly
what's wrong with it, how to fix it, and how to reconnect it to the genuine maths
that the Numberphile professors had in mind originally. Lots of amazing
maths look forward: non-standard summation methods for divergent series,
the eta function, a very well-behaved sister of the Zeta function, the gist of
analytic continuation in simple words, some more of Euler's mathemagical tricks etc. Now, I've tried to make this whole thing self-contained. So you don't have to have
watched my very different other video on one plus two plus three from over a year
ago or anything else to understand this one. Okay, let's get going. So that we are
all on the same page, here real quick is the whole Numberphile calculation.
They call the unknown value of the infinite series 1 + 2 + 3+... S. As
stepping-stones for the calculation they first calculate the sums of these other
two infinite series. So 1-1+1-1+... and 1-2+3-4+... Adding up the terms of the first series, we get the partial sums
1. Ok 1 minus 1 is 0, 1 minus 1 plus 1 is 1, 1 minus 1 plus 1 minus 1 is 0 and so
on. These partial sums alternate between 0 and 1 and so the Numberphile guys
declare that the sum of this infinite series is CLEARLY the average of 1 and 0
which is 1/2 (Marty) That's not all that clear to me. (Mathologer) Alright we'll get to that. They also mention that there
are other ways to justify this. We'll also get to that. Ok, second sum. Here they
start by considering what happens when you double this sum. So 2 times S2 is
equal to the infinite series added to itself but now before adding the two
infinite series they shift the bottom series one term to the right. Now 1 plus
nothing is 1, minus 2 plus 1 is minus 1, 3 minus 2 is 1, minus 4 plus 3 is minus 1,
etc. But that bottom series is the one we already looked at which, remember, is
equal to 1/2 and so .... Second sum done, great. Now for last sum, that's the one we're really after. Here the Numberphile guys start by subtracting S2, the
sum they just figured out, from S. Now 1 minus 1 is 0, 2 minus minus 2 is plus 4, 3
minus 3 is 0, 4 minus minus 4 is 8, etc. The zeros don't matter so let's get rid
of them. Take out the common factor 4. Ah the yellow that's our 1+2+3+...
sum S again. Now solve for S, and my usual magic here, and we get -1/12.
And here the Numberphile guys take a bow. But, not so fast! !ll this is really
nonsense the way it was presented. In particular these three identities are
false. This means that if on any maths exam at any university on Earth you're
asked to evaluate the sums of these infinite series and you give the
Numberphile identities as your answer you will receive exactly 0 marks for
your answers. It's critical to realise that in mathematics we have a precise
definition which underpins the sum of an infinite series. Wherever you see
infinite series this definition and only this definition applies unless there are
some huge disclaimers to the contrary in flashing neon lights. Alright, now the
Numberphile guys did not include any such disclaimers and so they too should
get 0 marks for their effort. (Marty) Or maybe give them -1/12 marks. (Mathologer) Yeah I think I can agree with that. OK,
what's this definition and what are the answers that will get you full marks on
your maths exam. To evaluate the sum of an infinite series you calculate the
sequence of partial sums just like the Numberphile guys did at
the very beginning. Now if the sequence of partial sums levels off to a finite
number, that is, if the sequence converges, or if it explodes to plus infinity, or if
it explodes to minus infinity, then this limit is the sum
of the infinite series. If no such limit exists, then the infinite series does not
have a sum. That's it, that's the definition. So for the first Numberphile
series the sequence of partial sums alternates between 0 and 1 and therefore
does not have a limit. This means that this infinite series does not have a sum,
neither 1/2 nor anything else. This is the correct answer for your maths exam.
Alright, what about the other two infinite series? Hmm, well, in the case of
1 plus 2 plus 3 the partial sums explode to plus infinity and so the sum
of the series is infinity. For the infinite series in
the middle the partial sums explode in size, but neither just to infinity or
just to minus infinity, and so this series also does not have a sum. So these
are the answers that get you full marks. In many ways the most important
infinite series are those with a finite sum which have not featured here yet. So,
to give some perspective, here's a standard example, an infinite geometric
series: 1/2+1/4+1/8 and so on. Now here the partial sums exhibit a
really nice pattern and clearly they converge to 1. (Marty) Yeah, I think this one is clear. (Mathologer) This one is clear and so the sum of this infinite series is 1. Oh, before I forget, those
finite sum series are usually called convergent series and all the other
infinite series are called divergent series. Keeping this in mind
let's have another look at the Numberphile calculation. Here's the whole
thing at a glance. It's just a transcript of the writing on the brown paper in the
Numberphile video. Again, as it was presented by Numberphile all this is
nonsense and worth 0 marks. (Marty) Or less! (Mathologer) Or less :) THIS. IS. NOT. MATHEMATICS. Don't use it,
otherwise you'll burn in mathematical hell. Having said that there should be some method to this madness, right?
Those guys are smart! But if there is, then it's clear that the sums you see
here cannot possibly represent the usual sums, as about six million people have
been misled to believe by this video. Ok let's start by doing something that may
also seem a little bit crazy. At first glance, just for fun, and in denial of
reality, let's assume for a second that those three Numberphile series were
actually convergent, that is, all had a finite sum. Then all, ALL highlighted
arguments would be valid. This includes the termwise adding and subtracting of
series that was performed here, ... and here, ... and even the shifting to the right
before the addition that a lot of people view with suspicion. Why would all these
operations be ok if we were dealing with convergent series? Because summation of
convergent series is consistent with termwise addition and subtraction, and
shifting. Let me explain that too. There are differences between finite and
infinite sums. For example, infinite series sometimes don't have a sum
whereas finite sums always exist, rearrangement of the terms can change
the sum of convergent infinite series, etc. On the other hand, the sums of
convergent infinite series do share a lot of the properties of finite sums and
it's exactly these properties that make them so useful. Here the most important
three such properties. Let's say you have two convergent infinite series, okay<
with sums A and B. Then, by adding these two series termwise you get a new
infinite series. And now it's quite easy to prove that this new infinite series
is also convergent and that its sum is equal to A plus B, of course. And the
same stays true if you replace all the pluses with minuses. So termwise addition
and subtraction is consistent with summing convergent series. That's
property one. Property two. Multiplying the terms of a convergent series with sum A by a number, say five, gives a new infinite series. Again
it's really easy to see that the new series is convergent and that its sum
is five times A. So termwise multiplication by numbers is also
consistent with summing convergent infinite series. That's our second
property. Finally, property three. Shifting the terms of a convergent series with
sum A by one term is the same as adding a zero as the first term to our series,
like that. Obviously, the new series is still convergent and its sum is the same
as that of the original series. This also works the opposite way. Removing a zero
term at the front does not change the sum. Okay, so that's property three. Now we can use these three properties to build valid arguments, very much like in the
Numberphile video. Here's an example. This is the convergent series that I showed
you earlier. Remember its sum is 1. Now let's say we
did not know its sum or even whether this series is convergent or not. Then we
could argue in a legit way like this: Ok, well we don't know whether it's
convergent or not, but if it's convergent and it's sum is M (M for mystery number :) then because of the number multiplication property we get that half M is equal to
1/2 times 1/2 which equals to 1/4 plus 1/4 times 1/2 which is equal to 1/8 and
so on. But because of the shift property 1/2 M is also equal to this guy here
zero plus whatever. Now because of the addition and subtraction property we are
justified to subtract as in the Numberphile video. So, on the left side we get M
minus 1/2 M, that's 1/2 M, and on the right side we've got 1/2 minus zero
that's 1/2, and then everything else kind of goes away. In total we get M
is equal to 1. So our assumption that our mystery series is convergent with sum M
lets us conclude in a valid way that the only value that M can possibly
be is 1. Question: Does this prove that M is 1? No, and this is very important.
Because this argument starts with an assumption, to be able to conclude that
the sum of the series really is 1, you still have to somehow show that the
series we started with is actually convergent. Hmm so this sort of argument
gives you an idea of what to expect but it does not get you all the way. Anyway
let's see what happens when we unleash this sort of reasoning on the first
Numberphile series which we already know is NOT convergent. Well, so just for kicks,
let's assume that the 1-1+1 series was actually convergent with the
unknown sum S1. Then, because of the shift property, S1 would also be
equal to 0 plus all the other junk. Again, because of the addition property, we can
add on both sides S1 plus S1 that's 2 S1 and then we add termwise on the
right to get, well, 1 there and everything else cancels out as you can see. And so we
get this which is exactly what Numberphile said S1 should be. So the
plot thickens here, right? In fact, in some ways this line of reasoning would
have fitted in better with the rest of their argument than just plucking the
number 1/2 out of thin air. So let's have another look at the Numberphile
argument and let's fit in what we just did as the first step to justify why S1
should be 1/2. So here we go. This is the Numberphile argument. Let's round it off
by inserting the argument for S1 from just a moment ago. Here we go.
And so here is one way to rephrase the whole thing to make it into a valid
argument. IF, and this is a big, huge monstrous IF, these three infinite series
were convergent, THEN this whole argument would be valid and the sums of these three
infinite series would be exactly the numbers given by Numberphile. Great, but
of course we know that the assumption of this valid argument is false, that the
three infinite series are not convergent. So, yes, this argument is valid in itself, but what good is it if the assumption is starts
with is false? Well here's an idea. Since no divergent series has a finite
value attached to it, let's dream big. What if it was possible to
extend the notion of summing a convergent series, what if it was
possible to define a super sum. This super sum should have three key
properties: first of all, we don't want to lose anything so the super sum of a
convergent series should be the same as its normal sum, right? Then all divergent
infinite series should be super summable with finite values. And then the last
thing we want is that super summing, just like normal summing is consistent with
adding, shifting, etc. alright. Now such a super sum extension of the standard
infinite sum would be as fantastic as the extension of the real numbers to the
complex numbers, with all sorts of cool properties for a smaller world remaining
true in the bigger world and at the same time all sorts of new magic appearing in
the bigger world. In particular, because consistency made the argument over there
valid for convergent series, we would expect the argument to still be valid if we
replaced ordinary equality by equality with respect to super summing. Even
better, since every infinite series would have a super sum, we could get rid of the
if and so the whole Numberphile video could be saved by just saying that we
are super summing instead of just boring old summing. What a lovely dream :) (Marty) It's time to wake up! (Mathologer) Yes, sadly. Well, anyway, those of you who watched my last video on this topic know that there
is a super sum. However it only assigns a sum to some :) divergent series but not
to all. In particular, it sums the first two guys over there. To show how super
summing works let's apply it to our first annoying divergent series. Basically
super summing builds upon normal summing and then averaging out any bouncing
around of the partial sums. We start by calculating the
sequence of partial sums. If this sequence converges, then our super sum
equals our normal sum and no tricks are needed. However if the sequence of
partial sums does not converge, as in the case of this infinite series, then we
start with the trickery, building a second sequence out of the first. The Nth term
of this new sequence is just the average of the first N terms of the first
sequence. So, for our particular series for the first term of the new sequence
there's nothing to average, well the average of 1 is just 1. Ok, then the
average of 1 is 0 is 1/2, the average of 1 and 0 and 1 is 2/3, etc. Now, every second number here is 1/2 and the remaining numbers also converge to 1/2 and this
means that overall the second series converges to 1/2. And that means that our
infinite series super sums to 1/2, which is also the number that Numberphile gets.
Now, for other infinite series even the second sequence may not converge in
which case we generate a third sequence, again by averaging the second sequence.
If that doesn't work, then we generate a fourth, then a fifth, etc.
As long as any of those sequences converges, the infinite series under
discussion has the corresponding limit as super sum. For example, in the case of
the second Numberphile series, the first and second sequences diverge but the
third sequence converges to 1/4 which then is our super sum. This is also
what Numberphile gets and so everything is looking good. Until now, now we have to
confront the sad truth that for most infinite series none of the associative
averaging sequences of numbers converge and so these series don't have a
super sum. In particular, for 1 plus 2 plus 3 and so on all the partial sums
are positive and obviously averaging over positive numbers will always only
result in positive numbers. In fact, all the associated sequences of
numbers will explore to positive infinity and so 1 + 2 + 3 etc definitely
does not super sum to anything finite, let alone anything finite and negative
like -1/12. Returning to the Numberphile calculation, here's the part
that can be totally justified using super sums instead of regular sums.
Because not every divergent series has a super sum we still need the big IF to
make this part of the argument valid. In itself not too bad, though, since the
assumption is actually true, right? And since super summing is really the most
natural extension of normal summing, 1/2 and 1/4 are the only reasonable numbers
to associate with the first two series. Really nice stuff I think. To recap, we
now know finite sums, convergen infinite sums and our new super sums. Oh,
by the way, I should mention that in the literature our super sum would be called
generalized Cesaro summation or generalized Hölder summation. Anyway,
these summation methods are proper extensions of each other and are
therefore able to assign meaningful values to larger and larger classes of
series. However, being able to do more also comes at a price. The more powerful
a summation method, the less well behaved it is. What works for finite sums cannot
necessarily be taken for granted for infinite sums. I already mentioned problems with rearranging convergent
series, for example. Of course, super sums also lack
all the nice summy properties that normal infinite sums lack but they are
even summy things that still work for infinite sums that no longer work for
our super sums. Yes, yes the three basic properties I've
been hammering are fine but to assume that any familiar summy property will
also work for super sums in general is risking zero marks (Marty): Or less! (Mathologer) Or less :)
For example, inserting or deleting infinitely zeros has no effect
on convergent series. However, doing the same to super sums can change things
dramatically. For example, if we insert infinitely many zeros into our first
series, like this, the super sum of the new series will be different from 1. Little puzzle for you: What's the new super sum? As usual, give
your answers in the comments. And this zero problem is important. I glossed over
this because it won't have any bearing on our discussion, but at some point in
the Numberphile calculation they simply zap infinitely many zeros and this
cannot be justified with our three properties. Bad... The effect of losing more
and more properties as you go more and more general is actually something that
you've all encountered before when you got introduced to larger and larger
number systems: fFrst the positive numbers, then to the integers, the
rational numbers, the reals and to the complex numbers and even beyond to the
quarternions and the octonions. Each time you broaden your world, you lose
some nice properties. Second puzzle for you: Can you think of some properties
that get lost along the way as we build larger and larger number systems? And
another puzzle: Suppose we assume that the 1 + 2 + 3 etc series actually super
sums to a finite number, can you manipulate this identity into a couple
of contradictory statements using our favourite three properties?
What can you then conclude from the fact that you can arrive at contradictory
statements? It will be interesting to see what you can come up with in this
respect? Anyway, it's time! (drum roll) We have to get serious about the
connection between our 1 + 2 + 3 series and - 1/12. So press the pause button,
go get your popcorn and your hot chocolate and let me know when you're
back :) (jeopardy music) Ready? Here we go. Even at the level of super sums we are
pretty far removed from what most people think of as a sum. After all for
divergent series, given all the averaging that is going on, the super summing is
really more like finding a super average than a real summy sum, don't you think? Well it will get more extreme not only in this respect but also in terms of the maths
that is required to understand what is going on with the -1/12
connection. I'm sure that a lot of you will already have heard of this
connection, so let me just state it first and then really explain it using the
Numberphile calculation as a template. This is the mega famous Riemann zeta
function. It is a function of the complex variable z. Written as the infinite sum
there it makes sense if the real part of z is greater than 1. However, there is one
unique way to extend the Riemann zeta function to an analytic function for all
complex numbers z excepting 1. Formally, if you substitute z = -1
you get ... Well, of course, minus 1 is not greater than 1, and so we really
don't have equality here, and so let's quickly get rid of their equal sign. Ok,
at the same time the right-hand side is our master villain 1 + 2 + 3 etc and the
value of zeta at minus 1 is equal to you guessed it -
1/12. And this, in a nutshell, is the genuine, real, actual connection
between 1 + 2 + 3 + and - 1/12. But why would anybody describe this connection as a
sum, and what has all this to do wit the last part of the Numberphile calculation.
Well there's more to explain. First, here's a mini introduction to analytic
functions and analytic continuation. This will be a rough and ready intro which is
all we need. You all know what a polynomial is, right? One of these guys:
a constant function or a linear function or a quadratic function or a cubic, etc.
Now let's play a game. Here is a chunk of a mystery continuous function that is
defined for all real numbers. So I'm just not showing you the part to the left of
the y-axis. Here's the question. Just by looking at this chunk, can you continue
the graph and tell me what my function is? Now you might be tempted to say 'Yes' but the answer is 'No'. There are infinitely many ways we could continue to the left
and here are a couple of examples. Here's one and there's another one, there's a third one, there's infinitely many different ones. Were you tricked? (Marty) No. (Mathologer) Sure you were not , but you know the game, right? Of course our initial chunk is part of a line and it's
natural to think of continuing the function as this same line. But we don't
have to. However, if I tell you the mystery function is linear, then your
initial chunk tells you exactly how to continue the function, there's only one
way to continue so that the whole function is linear. In fact, the same is
true if I only told you the function was a polynomial. Just by looking at the
chunk you could be absolutely sure that my polynomial is linear and exactly
which linear function, right? We can now generalize this simple observation a
couple of steps, in a pretty dramatic way actually. Here we go.
First, suppose that our initial chunk is part of a parabola, or if you like a
cubic, or any polynomial. If I then tell you that my mystery function is a
polynomial there's always only going to be exactly one polynomial that continues
our beginning chunk. In other words, a polynomial is completely determined by
any part of it. Second, all we've said stays true if we think of polynomials as
functions of a complex variable and if you begin with a chunk of the polynomial
corresponding to a region in the complex plane. So on the left, you see the complex
number plane where each point stands for a complex number
and I've also colored a small region in the plane. And so in terms of this
picture a polynomial is completely pinned down by the values it takes on
over a region like this. No other polynomial will take on all the same
values there. Again, just relax if all this seems a little bit too much.
Now, the polynomials are the simplest and most nicely behaved functions but there
is a much larger world of functions that shares a lot of the nicest properties
with polynomials. Those are the so-called analytic functions. These are the complex
functions that can be expressed locally as either regular finite polynomials or
as infinite polynomials, so called power series. For example, the exponential
function is an analytic function because it can be written as an infinite
polynomial like this. In fact, pretty much all our favorite functions such as the
trig functions, rational functions, etc. are analytic. Important for us is that
just like a polynomial, an analytic function is completely pinned down by
any initial chunk. So if I give you a beginning chunk of an analytic function
like the exponential function, then no other analytic function can continue
this chunk. This is usually expressed by saying that analytic continuation of an
analytic function is uniquely determined. In summary, though there may be many nice ad hoc ways to continue an analytic function there's just one distinguished,
most reasonable, absolutely fantastic never to be improved way to do this,
leading to a larger analytic function. Of course, as I said, this is all very
sketchy and you guys in the know will probably nitpick me to death in the
comments. (Marty) Looking forward to it. Ok, in particular I didn't tell you why we need to drag complex
functions into the discussion, but please just run with it for today and I promise
I'll fill in the details soon. In the meantime you can also read up on things
by following the links in the description. All you really have to
remember is this: an initial chunk of an analytic function nails down the whole
analytic function. Now we can join the dots. We have two completely different
notions of best extension. First, for extending sums to super sums of
divergent series and second for extending a chunk of an
analytic function to the whole analytic function. Combining these two extension
ideas, we can finally explain what's going on with 1 + 2 + 3 + and -1/12. Okay,
have a look at this infinite series. Notice that it's the same as the zeta
function except that it includes minuses. It's also obviously different from the
Numberphile series in that it includes a variable z. So it is actually an infinite
family of infinite series, one for each complex number z. Let's just make a
little list of such series corresponding to a few prominent integer values. For
z = 0 we our 1 minus 1 plus 1 series. Ok for x = - 1 we get 1 minus 2 plus 3 and Mathologer regulars know that for z = 1 and 2 the series are convergent. Now, in general, these series
are convergent for all complex numbers z in the positive brownish half plane. The
infinite series are divergent for all other z including 1 minus 1 plus 1 etc
at 0 and the other guy. However, just like the two Numberphile series can be super
summed, the same is possible for every z, for all the divergent series in this
family. And this allows us to define a close relative of the zeta function, the
so-called Dirichlet eta function. And this
function turns out to be an analytic function. So to start with, standard
summation only gives us part of this analytic function for which two infinite
series converge, this part here. Now, most mathematicians will simply discount a
divergent series that pop up here as useless artifacts. Instead they will
construct the analytic continuation of the eta function by completely different
methods. These methods are very slick and ingenious. However, they provide very little intuition and insight into what's really
going on here. On the other hand, seeing that the most reasonable extension of an
analytic function that is defined on part of the
complex plane is actually given by the most reasonable way to assign generalised
sums to these supposedly useless divergent series just feels right to me
and leads the way to a more intuitive understanding of analytic continuation,
at least in this case. But now here's a great thought: we just used a generalised
sum to construct an analytic continuation, right? Let's turn the idea
around, let's use analytic continuation to identify candidates for a generalised
summation method. And this is exactly what happens in the case of 1 + 2 + 3
etc and - 1/12 and the zeta function. You get the zeta function when you replace
all the minuses in the eta function by pluses. Well you get part of it, the
brownish part. which is the part of the complex plane for which the infinite
series on the right converge. For all other z the resulting infinite series
are divergent and even super summing doesn't help for the white part on the
left. So the super summing trick for eta just doesn't work for zeta. The
trick to use is encoded in the finale of the Numberphile pseudo proof. That's the
brownish bit down there. Remember this part of the argument takes the sum S2
of the 1 - 2 + 3 series and spits out a sum for the 1 + 2 + 3 series formally
these two sums are just what you get when you let z equal to -1 in the
infinite series of the eta and zeta functions and actually the Numberphile
calculation is just a special instance of a calculation involving eta and
zeta. Let's be brave. Ignoring questions of legality, let's unleash exactly the
same calculation on eta and zeta. So instead of subtracting S2 from S, let's
subtract eta from zeta. Ready? ... Right, take out the common factor down there. That's zeta again in the brackets. Now let's solve for zeta. There my magic again okay that's really exactly the
same as the last part of the Numberphile calculation using zetas and eta instead
of the Numberphile series. Just as a check after substituting z = -1 this identity turns into this, and with S2 being 1/4 we get
this. Okay more magic and we're back to -1/12. But didn't we say that the
Numberphile computation was nonsense? (Marty) Yes, we did. (Mathologer) We definitely did. And it is, but some
magic happens with zetas and eta which saves our zetas eta calculation from
being nonsense and that is the magic of analytic continuation. Both the series
for zetas and eta are convergent for every z in this brownish region. This
means that for these values of z our calculation and the resulting equation
above are pure, correct, 100% approved bona fide mathematics. But, as well,
eta is defined and analytic for all z and the same is true for the denominator.
But then the right side, as a quotient of two functions that are analytic
everywhere, is itself defined and analytic everywhere except possibly at
the zeroes of the denominator. In fact, a closer look reveals that the whole
right-hand side is analytic everywhere except at z=1. Here comes the punchline and this punchline hinges on the chunks-pin-down-analytic-functions business that I've been going on about. You should really try to
understand this. Okay, so both the left side and the right side are analytic in
this part of the complex plane here. But since the right side is analytic
everywhere, because of our chunks-pin- down-analytic-functions property, the
right side has to be equal to the elusive analytic continuation of the zeta
function on the left that everybody is really interested, the analytic
continuation of the zetas function. So this identity is a real jewel as it gives
an explicit way to calculate any value of the Riemann zeta function,
analytic continuation and all via the eta function which, remember, we defined via
super summing. In other words, it actually makes sense to use this identity as a
definition for the zeta function which works for all z. For example, setting
z=0 we get this one here and eta of zero was just the super sum of
1 minus 1 plus 1 etc which is equal to 1/2 and so zeta of 0 is equal to
-1/2. Here just a couple more interesting values for zeta. Nice. The zeroes are
particularly interesting. In fact, it turns out that zetas has zeros at all
negative even integers, -2, -4, -6, and so on. These are the
so-called trivial zeroes of the zeta function. I'm sure if you made it this far you've also heard of the Riemann hypothesis
which is all about other zeros of the zeta function. It says that all other
zeros are situated on this blue line here and what's also really interesting
is that on the right hand side we actually don't have to super sum to
calculate the values of zetas on the blue line because just with ordinary
summing eta can be evaluated everywhere in this part of the complex
plane, this part here which includes this critical line. Lots and lots of other
interesting things one could say here but, well, we are here to wrap up this
whole 1 plus 2 plus 3 is equal to -1/12 business. Alright, ok, let's do it.
In the first instance it is really our identity up there that the Numberphile
video is attempting to capture and it's definitely tempting to express his
identity as a new generalized sum maybe like this. I've decorated the equal
sign with an R in honour of Ramanujan who seems to have been the first to think
this way (NOT Euler as many people think). In fact, in Ramanujan's notebook we can
find a calculation very similar to the one that's in the Numberphile video. Of
course, there's a huge difference between a monumental
genius quickly abbreviating all his complicated stuff in a personal note and
a YouTube video addressed to a general audience, right? What you see up there is
part of a general method called Ramanujan summation that assigns values
to all sorts of divergent series including the three Numberphile series.
An important aspect of infinite series which is often overlooked is the order
in which the terms are summed. We are not just adding an infinite set we are doing
so in a certain order and this has all sorts of important implications. So, for
example, no matter how we arrange the natural numbers into an infinite series
this infinite series will always diverge to plus infinity using standard
summation. However, how exactly, how fast, slow, regular, erratic it diverges to
infinity depends very much on the order of the terms in the series. The Ramanujan
sum lacks pretty much all the nice summy properties that we encountered
today. However, it manages to capture aspects of naturally ordered series and
it pops up in many other branches of the theory of divergent series in
addition to the one we talked about today. Check out some of the links in
the description, especially if you know a lot of maths the article by Terry Tao.
Also I'm planning another video on the so called Euler-Maclaurin summation
formula which establishes a powerful connection between sums and integrals
and which is the starting point for Ramanujan's sum. Just to whet your
appetite here's one closely related -1/12 fact. The nth partial sum of
our infinite series is N(N+1)/2 So, if we plug in 1, 2, 3, 4 etc. for N
the formula will spit out those partial sums 1, 3, 6, 10. Now let's replace N by x
and graph the resulting function. That's a quadratic with zeros at 0 and - 1.
Now the remarkable thing about this graph is that the signed area here is
equal to -1/12. And this is definitely no coincidence. Really amazing
stuff, don't you agree? And that's it, finally, for today (and now I go and kill myself :) and I promise the next video will be a LOT shorter.
Does anyone know if numberphile have responded?
you can find responses on their twitter account and on their personal accounts eg: https://twitter.com/DrTonyPadilla/status/952939848855379969
And here we find the battleground where physicists and mathematicians duke it out.
So, if I understand this correctly, the sum of all natural numbers IS in fact strictly equal to -1/12, correct? It's just that the proof that Numberphile uses is just not necessarily a correct proof?