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if I ask you Brady what's one plus two plus three plus four and if I just keep adding forever what would you say the answer to that would be well I would say it would go get ten towards infinity yeah that makes sense doesn't it yeah am i right no it's minus 1/12 it's negative I've added all these positive numbers together up to infinity and I've got minus 1/12 and what's even more bizarre is that this result is used in many areas of physics in fact this result is critical to getting the 26 dimensions of string theory to pop out this result is critical in understanding the Casimir force in quantum electrodynamics but I thought what I would do today is we'd show a proof of this result that would be used to do due to Euler it gets a little hairy but if we just hang in there we'll then we'll do it together we'll go slowly so so the way it begins with is to rather than use these 1 plus 2 plus 3 we use let's just think of this series 1 plus X plus x squared plus X cubed and just keep adding higher and higher powers of X and what we know is that this is equal to 1 over 1 minus X that's a known result in fact it's strictly true for X less than unity so as it stands this is this is it's limited formally limited in its value we're going to we're going to manipulate it a little bit we're going to push the boundaries at which we were going to use it we can either prove that now or or if you want to accept it then let's accept this except with it's no accepting that it's not a difficult proof but we can accept it now we're going to do something which many of the numberphile viewers will know about some may not know we're going to do something called we're going to differentiate this and what that means is we're work out that what differentiation is is working out the rate of change so for example if you know the distance as a function of time and I differentiate it it tells me the rate of change of the distance or the velocity so the rule that you need to know for this is that the differentiation operator is called D by DX if it acts on something like X to the end and the only thing we need to know is that this gives me brings down the N and it lowers the power by one that's all we need to know in order to be able to carry on so we do it well the differential of one is zero the differential of X this is X to the 1 so n is 1 so this brings down a 1 and that becomes X to the 0 which is 1 so the first term is 1 X Squared's well I differentiate X squares for N equals 2 here so it becomes 2 times X the next term is from X cubed so n is 3 so this becomes 3 x squared ok so you get the idea the next term would be 4 x cubed for example and this equals and you'll have to just accept my word for it at the moment the right hand side when I differentiate it we just becomes 1 over 1 minus x squared now what I'm going to do is I'm going to set X to be equal to minus 1 all right this is what I did you just do it what do you mean you just going to do that yeah well I'm just going to fit it into this see what it tells me this form is going to give me so what does it give me it gives me 1 minus 2 because of X is minus 1 plus 3 because X x squared is positive minus 4 plus 5 etc - and if I set X equal to minus 1 here it becomes 1 over 2 squared so this is 1/4 so this is already beginning to look a little strange hang on this seems significantly so what are ways we've just shown that 1 minus 2 plus 3 minus 4 plus 5 minus 6 plus 7 is equal the way up to infinity is equal to 1/4 and now going to introduce I think one of the best functions ever okay and I'm sure we're going to be seeing a lot of it in numberphile over the next few months maybe and it was first introduced I think by Euler and then it was generalized by Riemann and it plays a huge role in understanding the distribution of prime numbers and it will play a huge role in verifying this calculation okay let's go it's called the Riemann zeta function sometimes called the Euler Riemann zeta function and it's definition is the following it's got a funny symbol so it's got a symbol Zita and the argument of it is called s and its definition is it's the infinite sum so I'm going to sum from of the integers from N equals 1 to infinity of 1 over N to the S ok so what what is this this is 1 well let me just 1 over 1 to the S well that's just 1 plus 1 over 2 to the S plus 1 over 3 to the S plus 1 over 4 etc now euler he said that s was real right it was a it was a regular number along the real axis what riemann did and I'll just quote it here for now is that he allowed s to be made up of a real part which we'll call Sigma plus an imaginary part which will call T so this is a complex ok so this is this was the difference between Euler's approach and riemann riemann generalized euler you can see sigma is the real bit and then T is the complex bit this is called and it's an imaginary number because it's I times T and together they form a complex number we're going to concentrate on the real bit here what oil are now did he multiplies this by 2 to the minus s times Zeta of s ok so what is that so this is just 2 to the minus s plus another way of writing 1 over 2 to the S is 2 to the minus s so this is equal to plus 4 to the minus S Plus this is here this is 3 to the minus s times 2 to the minus s gives me 6 to the minus s plus 8 to the minus s plus all the way up ok so he just multiplied this Zeta by by this so what oiler now did was he it takes two of these away from this ok ok so let me just write it out slowly so and let me just write out the first term which is the Zeta so it's 1 plus 2 to the minus s plus 3 to the minus s plus 4 to the minus s plus etc and now if I take away this twice this thing I get the first thing I get is I'm taking away - to lots of and I'm going to line these up ok I'm lining the two up there I'm going to line the four up there and then I would line the six up you know with the matching the six oh but now what you can see so the 1 just lives by itself that's ok I've got 2 to the minus s minus 2 lots of them so this is gives me minus 2 to the minus s the 3 is by itself so that just stays as it is 4 minus 2 lots of 4 gives me minus 1 lot of 4 so you can see I'm beginning to alternate ok and I need some more paper Brady now so now what he does as he did above there when he set X equal to minus 1 he sets s equal to minus 1 ok and now let's just ask what what does this expression look like so on the left hand side here when we do that we've got 1 minus 2 times 2 to the minus - minus 1 which is plus 1 so it's 1 minus 4 which is minus 3 okay but what is Zeta of minus 1 well if I look up here and put s equals 2 minus 1 in here can you see that this becomes 1 plus 2 Plus 3 plus 4 etc so all the way up forever and now on the right hand side so I've done the left hand side on the right hand side I do the same I set s equal to minus 1 and you can see what I've got here is 1 minus 2 plus 3 minus 4 plus 5 etc but he's already shown what that is if we look here we've already seen that 1 minus 2 plus 3 minus 4 plus 5 is 1/4 and I divide out by the 3 minus 3 and I get the 1 plus 2 Plus 3 plus 4 plus is actually equal to minus 1/12 and that is the Euler proof in quotation marks that the sum of the integers is minus 1/12 and the so it sounds crazy right it does of it it sounds crazy and what what you can do and what Riemann was able to show is that this actually makes sense when you look at the riemann function the Riemann zeta function the complex version of it and set s equals to minus 1 he showed that zeta of minus 1 indeed is equal to minus 1/12 and I think a way of understanding that what's going on here is that when you look at it in terms of the Riemann zeta-function there is actually it diverges at one point the Riemann zeta-function and that one point is when s equals one so Zeta so remember what Zeta and says Zeta of s we said was 1 plus 1 over 2 to the S plus 1 over 3 to the s etc so if I set s equal to 1 then Zeta of 1 is 1 plus 1/2 plus 1/3 plus 1/4 and this is a very slowly diverging series this tends to infinity and I think one way of kind of trying to understand why it is that the the actual Riemann zeta function which we may describe in more detail when we come to think about the primes why Zeta of minus 1 is equal to minus 1/12 the reason why this is is consistent with the fact that we actually think when we sum over the integers they're actually infinite is that in in some way we're having to go through this point here and what Riemann is able to demonstrate is by kind of circumventing it which is the the physical way of saying removing the divergence we can end up with these sensible numbers like minus a12 so professor last I mean the question I have is if I sat down with a calculator and wrote 1 plus 2 Plus 3 plus 4 plus 5 who did it forever and then press the equal sign am I going to get minus a12 you see oh I don't know in reality one of the things you have to be careful about here that we have to be careful about is that what you have just stated there is basically using the set of axioms that way that were used to in in this in in the context of finite sums these are finite that these are infinite sums so in other words you could you could keep going to the end of the universe typing in the numbers and you still have made epsilon contribution to this overall sum and any desire means really really small and I think that hidden in there is a subtlety of death you've got to deal with divergent numbers divergent sums very carefully and so the logic which goes in to show that you know something is clearly divergent and as you were just saying there with it with the putting it into the calculator doesn't necessarily follow when you're really dealing with these divergent sums you have to be very careful as to understand where the divergence is emerging and I think this is an example of a case where this Riemann zeta-function is perfectly well behaved at 0 of -1 and it's it's it's it's called an analytic continuation it's taken you from one regime where it clearly looks like it's divergent and moving it into a regime where it's better to find I can accept that zero is a number of some significance and that one is and that infinity although it's not numbers a significant thing and I can accept PI as you know I and II but minus 1/12 just seems so oh yeah I know but but it does player a play a role in lots of different things does that the number 12 and and it's so for example as I said the the calculations of the critical dimension in string theory that 26 dimensions comes from this calculation in fact is basically the strings live in a two-dimensional moving a two-dimensional plane and and then they've got orthogonal directions and there's 24 of these orthogonal directions which is 2 times 12 and and that's this calculation is where and you actually get the critical number of dimensions it gives me the 20 before and there's a wonderful mathematician called John Bayes or by Bayes let me just write him down and people might want to go and look up some of his work he's he's given seminars on the importance of twenty four and eight and it's used in all sorts of different packing formulas it's it appears all over first let's let's go back and look at this and why is this answer seemingly on one level ridiculous well let's look at the the first n M integers at first n natural numbers so we actually do the sum what 1 to N so you go 1 plus 2 plus 3 all the way up to n okay right now we know the answer to this it's n n plus 1 divided by 2 then what the emphasize here is we this is called this here that I've written down here is that is the partial sum this is the nth partial sort of clearly as I take n to infinity this numbers just getting bigger and bigger and bigger and bigger I should be getting the answer infinity but of course you know infinity doesn't really exist in nature does it you know you don't measure an infinity in nature you just don't do that but you can attach a value to this sum you can attach it and number two this summon its unique and there's various ways in which you can do that one way is a this is a special case of the Riemann zeta function so what is the Riemann zeta function the Riemann zeta function is the following define like so it's the sum from N equals 1 to infinity of n to the minus Z ok so if Z is bigger than 1 this one is perfectly well defined it's a nice finite answer so you're basically just taking numbers they get smaller and smaller and smaller and smaller and adding them together ok it's intuitively feasible you would get a finite answer from that so from Z greater than 1 you get this now what you do is you do something very important you do this trick of analytic continuation that's what it's called basically you know that you have you have this this formula so you have some expression for this sum that works for Z greater than 1 and then you say ok let's assume that that expression also works for different values of Z where it's not greater than 1 where it's less than one thing and actually assuring that the the functions that you're talking about have very special properties namely that our analytic that means that all their their continuous functions that they're nice and smooth there's no kinks or anything like that assuming that holds and you can make this extension when you take the analytic continuation of the Riemann zeta-function and you evaluate it 2-1 I chose minus 1 because they plug in minus 1 I just get the sum from N equals 1 to infinity of n which is precisely this sum here ok so at minus 1 well I do the analytic continuation I plug it into my nice formula that I derived for Z greater than 1 buying our plates is there less than less than 1 and I get minus or 12 okay through this process of analytic continuation now you're probably still skeptical not really believing that this works but but we can prove it without doing this ok just to show that we can there are ways to get to these loads of ways you can get to this answer ok this is the sort of the classic way using this Riemann zeta-function an analytic continuation but there are a million ways to do is okay so shall we just prove it yeah let's just this it's socially more paper yeah when in physics do you need to know what 1 plus 2 Plus 3 plus 4 plus 5 plus 6 plus 7 etc well ok so so essentially in physics what you do is in okay so what we saw one example rate in impulsively string theory book where he was trying to predict the dimensionality of the universe and actually this sum appears because you end up having to sum up all different you can think of a string right actual the strings that build string theory are built from they have different modes of oscillation they have the sort of single one first harmonic second harmonic third harmonic and so on and so on is you have an infinite number of harmonics and you have to sum up all their contributions and when you sum up all their contributions in naively you would get an infant answer that actually no you don't you get minus 1/12 a period and that's the actually the minus 1/12 and then feeds back into 26 and dimensions but it's not just in string theory it's it these kinds of songs which naively appear and integrals as well integral is just a continuous sum these kind of divergent song divergent integrals they appear all over all over quantum field theory where you have seemingly infinite contributions but you can regulate them you can make sense of that infinity and there and the answers how you make sense that infinity should not affect the answer and indeed it doesn't and the certain properties of sums that that you can that you can they have to satisfy certain theorems and whatnot that so that they do make sense that they do give unique answer some songs just are too crazy but certainly ones like this a totally totally nice and you can get nice finite answer that