(Spoiler alert: Yes it does . . . probably . . . um, maybe . . .I mean, it has to.)

An interwebs firestorm has been raging recently about a Numberphile video that makes the astounding claim that if you add up all the positive whole numbers from one to infinity, the result will be -1/12. To write it out more concisely

1+2+3+4+ . . . = -1/12 , (where the three dots indicate all the rest of the positive numbers up to infinity)

If you haven't seen the video, take a look - it's short.

Fascinating, isn't it?

Renowned science writer and astronomer Phil Plait (Bad Astronomy) blogged about the video recently, calling it "simply the most astonishing math you'll ever see." The post led to a Twitter and comment storm, fueled both by people bowled over by the calculation and a much larger number of people convinced it was nothing short of mathematical fraud.The passionate response he got to his post led Plait to write a follow up piece, partly in self defense, and partly as penance for his various mathematical sins as pointed out by his readers.

Clearly, only a fool would consider defending this absurd calculation after the reception Plait got.

So here I go . . .

**First, About the Math**

I'm not going to follow Plait's example of trying to explain the math that goes into the calculation. But I will point out that many of the problems that commenters and Twitterers latched onto are irrelevant if you look at the more elaborate discussion in the Numberphile's extra footage. In the much longer second video, they come to the same reviled result as in the first video, except that they use an approach first written down by Leonhard Euler.

If you dislike the initial video, you really should watch this one to see if it sways you at all.

That's much better, isn't it? I'm sure it's not perfect, but the flaws are beyond my mathematical abilities to recognize.

In any case I'm willing to believe 1+2+3+4+ . . . = -1/12 is a mathematically legitimate thing to write down for the following three reasons.

1. Euler, who was one of the greatest mathematicians of all time, proved the equation for real numbers.

2. Another great mathematician, Bernhard Riemann, generalized Euler's approach to include complex numbers, and came up with the same equation.

3. My favorite mathematician, the self-taught genius Srinivasa Ramanujan, rediscovered the equation and stood by it, even though he realized that he might be thought be mad for making the claim, writing in a letter to mathematician G.H. Hardy, "I told him that the sum of an infinite number of terms of the series: 1 + 2 + 3 + 4 + · · · = −1/12 under my theory. If I tell you this you will at once point out to me the lunatic asylum as my goal."

So, counting the Numberphiles' somewhat dubious derivation, there are at least four ways to prove that the sum of all the positive integers equals -1/12. And as far as I know, there's no way to prove that it doesn't equal -1/12.*

If you don't believe any of these people, then there's nothing I can do, mathematically speaking, to change your mind. I mean these guys are among the greatest. What could I add that would improve on their proofs?

**But, Obviously ****1+2+3+4+ . . . = -1/12 Doesn't Really Mean Anything - Right? **

Of the mathematicians and physicists I've talked to about it, several of them are willing to accept that it's possible to derive the equation, but insist that it's meaningless. They tell me, if I understand them correctly, that it's some sort of numeric fluke that can't possibly have any consequences in the real world. There's just no way to add positive numbers together get a negative result in reality, especially when the numbers you're adding are getting larger and larger. In effect, it's nothing more than an artifact that results from a method that makes sense when applied to complex variables or other series, but not for the sum of positive integers. To think otherwise would be nuts, right?

The problem is, they're wrong (or so a number of physicists have told me). The equation 1+2+3+4+ . . . = -1/12 is vital for describing the real world.

As the Numberphile people point out, the dreaded equation pops up in many places in physics. They specifically note it's appearance in a string theory textbook (see page 22 in this Google book). But that's only one example and, depending on how you feel about string theory, among the least convincing ones. What's much more compelling is the fact that this sort of equation is integral to Quantum Electrodynamics (QED).

QED is the theory that explains the interaction between charged particles like electrons and protons. Along with neutrons, electrons and protons make up atoms, which in turn make up molecules and everything built of them. In other words, QED essentially describes much of the physical world we live in. And it does it extremely well. QED calculations for the spin of the electron have been confirmed to better than one part in ten trillion - making QED just about the most precise and successful theory of all time.

If QED is correct (and it appears to be the most correct theory yet developed, if experimental confirmation is a reasonable way to judge correctness), then I would argue that the things that go into QED calculations must be just as correct. Doing QED calculations requires using 1+2+3+4+ . . . = -1/12, so the equation is at least as correct as QED theory itself.

In fact, the Wikipedia page on a QED phenomenon known as the Casimir Effect shows a derivation of the effect that includes an even more audacious equation involving the sum of the *cubes* of the natural numbers up to infinity. Specifically, calculating the effect involves using the equation

1^3+2^3+3^3 +4^3+ . . . = 1/120, (where the notation 2^3 means 2x2x2)

(In the Wikipedia article, they have an equation that looks like this , but the stuff on the left hand side is just another way of writing 1^3+2^3+3^3 +4^3+ . . .)

The number on the right is positive this time, but it's ten times smaller than 1/12, even though each of the terms in the sum is much bigger than the corresponding terms in the equation 1 + 2 + 3 + 4 + . . . = −1/12 (except for the first term, of course, since 1^3 = 1). Both equations come from the same sort of derivation, so it's not surprising that they are both seemingly incredible and ridiculous. But if you believe in QED and the Casimir Effect, how can you not believe the pieces that go into them?

**Maybe It's Just a Trick**

One response I've gotten after querying my more mathematically savvy friends is that the equations are nifty tricks, and nothing more, to get rid of infinities in QED and produce the correct finite answers. I guess that's possible, but you would have to be one heck of a mathamagician to come up with a trick resulting in accuracy of a part in ten trillion.

It's even more impressive when you consider that the QED predictions came before the experiments that measured things like the electron spin to fourteen decimal places. It's one thing to design a trick to rationalize a number you already know. It's a whole other matter to come up with a trick that gives you the answers in advance of the experiment. In that case, it's not a trick, it's simply a very good theory.

**Maybe It's Not Necessary, Just Handy**

One final possibility that I can think of is that the equations are not really necessary for doing QED calculations, and that instead there is a correct and intelligible approach that gives answers without using nonsense like 1 + 2 + 3 + 4 + . . . = −1/12 or 1^3+2^3+3^3 +4^3+ . . . = 1/120.

I can't imagine why physicists would rather rely on trickery than doing things correctly, so I tend to dismiss the idea that some sort of mathematical conspiracy is behind it all. If it turns out that it's possible to have physical theories that describe the real world as well as QED does without relying these equations, then we might as well use those theories and forget the whole controversy.

**So What's Really Wrong?**

If you accept that Euler, Riemann, and Ramanujan did things properly when they found 1 + 2 + 3 + 4 + . . . = −1/12, and if you accept that it and related equations are necessary to describe the real world, then how can you not accept that the equation is true? And yet, many people still claim that there's something wrong. It doesn't make sense. It's so counter intuitive that the phrase "counter intuitive" seems far too weak a description. It's an alien, freakish, mind f----.

But that's OK. Some things are true without being conceivable. This is just the most recent example I can think of. Pythagoras and and his followers apparently committed human sacrifice because they couldn't handle the idea of irrational numbers. For centuries, ancient mathematicians struggled with unsolvable problems because they didn't know that pi is a transcendental number. And today, there are still things about quantum mechanics that defy intuitive understanding - the whole point of Schrodinger's Cat is to illustrate the absurdity of quantum superposition. But just because people didn't intuitively grasp those things, it didn't change the fact the the square root of 2 is irrational, that pi's transcendental nature means it's impossible to square the circle, and that particles can become quantum mechanically entangled just like Schrodinger's Cat.

Yes, there's a problem with 1 + 2 + 3 + 4 + . . . = −1/12. But I suspect the problem is with us and our failure to understand infinity. Why shouldn't an infinite sum of numbers going to infinity add up to a finite (and negative!) number? I don't really know what infinity means anyway, so I can't think of any way to object to a statement that includes not one but TWO infinities in it.

You might as well ask me why a bandersnatch of numbers going to bandersnatch add up to -1/12. But if you're able to mathematically sum a bandersnatch of bandersnatches, and then use that sum to describe the real world and predict the outcomes of real world experiments ~~I have no choice but~~ it seems unreasonable not to believe your bandersnatch math.

**So, Does 1 + 2 + 3 + 4 + . . . = −1/12 or not?**

You bet your bandersnatch it does! (I think, anyway)

## Wednesday, January 22, 2014

### Does 1+2+3+4+ . . . =-1/12?

Posted by Buzz Skyline at 1/22/2014 03:44:00 PM

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But analytic continuation *is* a trick. It's specifically a trick to extend the values of certain types of functions outside the domain where they are valid. A summation of the series of natural numbers isn't valid -- it diverges.

ReplyDeleteSumming any number of natural numbers can never produce a nonpositive number or a fraction. Infinity just sums an arbitrarily large quantity of these numbers, so axiomatically 1+2+3+...+n can't ever equal anything but an arbitrarily large positive number.

Critically, analytic continuation extends a function beyond its definition, so the summation of the natural numbers is no longer strictly speaking what we're talking about, just like tangent doesn't actually have an inverse function, which doesn't make arctan less useful.

So the question that boggles our ape brains is what analytic continuation is really doing when it extends functions into the unknown realms where they are not defined, other than producing useful answers for equations in which they're involved.

Does it have any meaning in the real world? Who knows? What does that question even mean? Does infinity even have any meaning in the real world, or is it just a handy trick for calculating the area under a curve?

I don't understand what you mean by calling analytic continuation a trick. According to Wikipedia, it's a mathematical technique. What defines a mathematical trick?

DeleteIt is a trick for a simple reason - zeta function is only defined for s>1, whereas to perform the trick of "summing up" the natural numbers if set s=-1. This may be a useful trick, but it's still a trick.

DeleteI was unsure why the numbers of S2 were shifted to the right in order to get the desired result?

ReplyDeleteI assume you're talking about the first video. One of the problems with it is the inappropriate manipulation of series. It's wise of you to question that step. But rather than try to rationalize it, I would recommend you watch the second video and just forget about the first one.

DeleteYou cite the Wikipedia page of the Casimir effect. In fact, this Wikipedia derivation (http://en.wikipedia.org/wiki/Casimir_effect, part "Derivation of Casimir effect assuming zeta-regularization", version from 13 January 2014, 20:07) is equivalent to the problem discussed here and clearly shows the most important point: Regularization! The Wikipedia article does not merely sum 1^3 + 2^3 + 3^3 + 4^3 + ... = 1/120. Instead, it sums 1^(3-s) + 2^(3-s) + 3^(3-s) + 4^(3-s) + ..., where s is a (complex) regularization parameter. In the end, s=0 is needed, but already at an early point of the calculation, this parameter s is introduced in order to make the calculational steps mathematically well-defined. The sum which appears only converges for s>4 (or, rather, the real part of s must satisfy Re(s)>4). Under this condition, the sum yields the Riemann zeta function, 1^(3-s) + 2^(3-s) + 3^(3-s) + 4^(3-s) + ... = zeta(s-3), for Re(s)>4. So we get the result for the Casimir energy as a function of the (complex) regulator s. This function is analytic for Re(s)>4, so it can be analytically continued to other values of s, including s=0 which is the desired value for s where the regularization disappears. This function value, zeta(-3) = 1/120, is known, it can be derived by techniques from complex analysis. But zeta(-3) is *not* given by the divergent sum 1^3 + 2^3 + 3^3 + 4^3 + ... which has no well-defined value. Only when specifying a proper regularization scheme with some parameter s, we can perform the calculation, evaluate the infinite sum (as a function of s), analytically continue its result to s=0 and obtain zeta(-3) = 1/120.

ReplyDeleteCan you explain this regularization?

Delete. . . "and obtain zeta(-3) = 1/120" where zeta(-3) = 1^3+2^3+3^3+ . . .

ReplyDeleteso 1^3+2^3+3^3+ . . . =1/120, as Buzz said.

No! We have zeta(-3) = 1/120, but zeta(-3) is NOT equal to 1^3 + 2^3 + 3^3 + ...! For argument -3, the zeta function cannot be defined through this sum which is one of its possible definitions for arguments larger than 1.

DeleteHave a look at possible definitions of zeta(z) e.g. in this book: I.S. Gradshteyn and I.M. Ryzhik, "Table of Integrals, Series, and Products" (I found a PDF online there: http://atsol.fis.ucv.cl/dariop/sites/atsol.fis.ucv.cl.dariop/files/Table_of_Integrals_Series_and_Products_Tablicy_Integralov_Summ_Rjadov_I_Proizvedennij_Engl._2.pdf).

Section 9.513 (page 1036) shows several integral definitions for zeta(z), some of them also valid for negative z, from which one may evaluate zeta(-3). However, the well-known series representation for zeta(z) in section 9.522 1. (page 1037) is clearly indicated to be only valid for Re(z)>1, so in particular z>1 on the real axis. So this sum cannot be used as a definition for zeta(-3).

In simple terms: Look at f(x) = 1/(1-x). This function is perfectly well-defined for x = 2, then f(2) = -1. On the other hand, for |x| < 1 we can write f(x) = 1 + x + x^2 + x^3 + ... But we cannot define f(x) through this infinite sum for x = 2, because 1 + 2 + 4 + 8 + ... is divergent and does not have a well-defined value. Only within the "radius of convergence" |x| < 1, we can define f(x) through 1 + x + x^2 + x^3 + ... There, however, we may evaluate the convergent sum and obtain f(x) = 1/(1-x). And by defining f(x) through 1/(1-x) we may extend its definition from |x| < 1 to all values of x except 1. Still, the sum 1 + 2 + 4 + 8 + ... is divergent and does not have a finite value, only f(2) does, using the definition f(x) = 1/(1-x).

Thanks for bringing some clarity into this horrible mess!

Delete1+2+4+... is a great example, because it does converge 2-adically, where |2|=0.5 < 1, and it converges to -1 = f(2).

ReplyDeleteA shadow of this fact is that the binary representation of -1 is 11...11 - calculating mod 2^n is a quotient of calculating 2-adically, and under this quotient the image of the sum is the finite sum of powers of 2 that gives 11...11, whereas the image of -1 is -1 of course, so they're equal.

Now this doesn't have a much to do with zeta functions afaik, but it is a good example of why a "crazy" thing to do with a sum can be made quite precise mathematically.

http://en.wikipedia.org/wiki/P-adic for more on p-adics, or the excellent book by Gouvea http://www.amazon.com/p-adic-Numbers-An-Introduction-Universitext/dp/3540629114/ref=tmm_pap_title_0?ie=UTF8&qid=1390554269&sr=8-1

You may claim that 1+2+4+8+... (or other sums) converge p-adically, but that is a completely different notion of absolute values and convergence which - as far as I have seen - nobody was using so far, neither in this blog nor in the Numberphile videos.

ReplyDeleteStill, it is an interesting phenomenon that 1+2+4+8+... converges 2-adically to the same value as the corresponding analytic continuation of the geometric series.

What I claim is that, under the "standard" definition of absolute values and numbers (which should be assumed unless specified otherwise) the above-mentioned infinite sums do not converge and are not equal to any finite values (-1, -1/12, 1/120 etc.).

Nobody said anything about the series converging. They don't converge. They are simply equal to -1/12 and 1/120 when you include infinite terms. But the corresponding series diverge. It's a paradox, and yet true

ReplyDeleteThe value of a convergent series is naturally and uniquely given by the value it converges to. But it is not obvious how to define a unique value for a divergent series!

ReplyDeleteLook at the people you cite above for providing 1+2+3+4+... = -1/12:

The Numberphiles' derivation (1st video) you call yourself "somewhat dubious", and there are so many flaws in it that I would not even dare to consider it a derivation.

Euler's proof (as cited in Baez's exercise sheet you linked above) seems to me roughly the same what Numberphile presents in their second, longer video. And Baez himself admits two things: 1) He used "dirty tricks invented by Euler". 2) He needs to define some divergent series by their "Abel sums". Because of 2) Euler's derivation as presented by Baez is dependent on a special definition (see below). And because of 1), especially the moment when the regularization parameter s disappears, this is again a step where expressions are used which are not rigorously defined and depend on a specific choice of regularization (here using this parameter s). In the end, Baez mentions the analytic continuation of the Riemann zeta function, but he never claims that zeta(-1) is equal to the sum of natural numbers (it is just the analytic continuation of a series with generic powers (-s)).

Also Ramanujan only claims this identity to be true only "under his theory".

Conclusion: In order to define a value for a divergent series, you first have to specify your particular theory or regularization scheme which makes your definition well-defined and consistent. And, because this is then dependent on your definition, how do you know that your value is unique? There are always several ways of defining things.

I might be tempted to define the value of any divergent series by summing up exactly the first 100 of its terms and stopping there. This definition would also be fine. For (fast) convergent series it already provides a decent approximation to their true value. And for divergent series the size of this value is some kind of measure for the degree of divergence. And surely, for all series mentioned here, this so-defined value differs from the one claimed by you (-1/12, 1/120 etc.).

The regularization via the Riemann zeta function or via the Abel sum mentioned in the Baez / Euler approach have in common the following scheme:

Given a divergent series F = f_1 + f_2 + f_3 + ..., you introduce a regulator s which changes the series into F(s) = f_1(s) + f_2(s) + f_3(s) + ... such that f_n(0) = f_n, i.e. your original series is recovered for s=0. The new series F(s) should converge within some region of s. Within this region, the series is summed up and its value F(s) as a function of the parameter s is obtained. This function F(s) (not the original divergent series) is then (analytically or by taking the limit) continued to s=0. Finally one defines the value of the original series F to be F(0).

For F = 1 + 2 + 3 + 4 + ..., we have f_n = n.

Using f_n(s) = n^(1-s), we obtain F(s) = zeta(s-1) and therefore define the original series' value to be F(0) = zeta(-1) = -1/12.

We could also have used the Abel sum here by f_n(s) = n*(1-s)^n. Then F(s) = (1-s)/s^2. Here the limit s->0 does not exist, so the Abel-sum definition fails or provides at best the "value" infinity, which is different from -1/12.

I want to stress the following: Especially if want to calculate something meaningful in physics, you should not start by some ill-defined expression, perform non-rigorous transformations on it and be happy to arrive at some finite result. This can go awfully wrong! Always start with an expression which is well-defined by introducing a decent regularization scheme from the beginning. Then perform well-defined transformations and only remove the regulator in the end, when the resulting expression is well-defined and finite without regulator.

ReplyDeleteThis is the way which my former colleagues from Quantum Field Theory (QED etc.) and I are using when we are dealing with regularization and renormalization schemes for evaluating loop integrals of Feynman diagrams etc.

Don't ge me wrong: I love "1+2+3+4+... = -1/12"! It's a funny equation, we were having it on the clock of my last physics institute's seminar room (in the form "-1/sum_{n=0}^\infty n" at the place of the "12"). But this equation is not a rigorous mathematical identity, it rather is a mathematical joke with a large portion of reality background.

While I agree that it's preferable to have a solid foundation before attempting to calculate something meaningful in physics, as a rule, I feel like it's too late to apply that standard to QED, among other things, when it comes to 1+2+3+4+... = -1/12.

ReplyDeleteIt's interesting to delve into the mechanics and details leading to Euler's, Reimann's and Ramanujan's derivations of the equation, but physicists have been using it to make extraordinary theories that describe the world very well. If the equation were different, or didn't exist at all, then the theories (so I'm told) wouldn't work.

So even though mathematical rigor seems to make the truth of the equation debatable, to mathematicians anyway, the very existence of a world that obeys the theories that depend on it, to a precision of 13 significant digits or more in some cases, is a pragmatic argument for the fundamental truth of 1+2+3+4+... = -1/12.

I hope mathematicians will eventually be able to discover why this is the case. For all I know, it may turn out to be just a fluke. But any joke that can tell me about the real world to a part in ten trillion seems like a very handy joke. Perhaps, as comedians like to say, "It's funny because it's true."

QED is a great theory which makes powerful and precise predictions. I know QED because as a theoretical particle physicist I have been working with QED and other quantum field theories for more than twelve years.

ReplyDeleteBut QED does NOT rely on "identities" like 1+2+3+4+... = -1/12. Of course, in QED we encounter infinities almost everywhere. But they are treated by employing regularization and renormalization schemes. In a calculation, if we arrive at a divergent series like 1+2+3+4+..., we know that we have to go back some steps and look for a way to properly regularize our expressions in order to work with well-defined quantities.

Again I refer to the Wikipedia page for the Casimir Effect: Instead of proceeding with a divergent expression, it introduces the regulator s which makes both the integrals and the series well-defined. Instead of 1^3 + 2^3 + 3^3 + ... it sums 1^(3-s) + 2^(3-s) + 3^(3-s) + ... = zeta(s-3), and in the zeta function (not in the series!) the regulator s can be removed because of analytic continuation.

Of course, physicists are pragmatic, and we cannot always prove mathematically that what we are doing is correct. Fortunately, physicists and mathematicians before us have already proven the correctness of many regularization procedures we are using today. Even weird schemes like dimensional regularization (where the number of space-time dimensions is changed from 3+1=4 to a generic complex number) have been proven, and people have elaborated which calculational transformations are valid there. This makes it relatively easy for us today to simply use such a scheme without having to bother each time about its correctness.

Let me show you a case where physicists have tried to perform a quantum-field-theory calculation without using regularization:

http://arxiv.org/abs/1108.5872

This calculation had been done before using dimensional regularization. The authors of the new paper did it without regularizing and got a wrong result! (Notice that their paper never got published in a journal.) The authors themselves pointed out the crucial integral, written down in equation (4.4). While the dimensionally regularized version of this integral yields a finite (and correct) result, they claimed this integral to be zero because of "symmetric integration" (i.e., rotational invariance). But their argument is not valid because the integral is not absolutely convergent and depends on the order in which the four components of the loop momentum are integrated. So even renowned physicists can easily make mistakes when they try to solve problems without properly regularizing their expressions.

To conclude: Using analytic continuation in combination with regularization is crucial for QED (and other fields of physics). But no physicist who wants to obtain correct results naively uses "identities" like 1+2+3+4+... = -1/12 without at least knowing what is the correct way of obtaining this result by employing a proper regularization. This "identity" as it stands here is simply wrong. But it can be useful for a calculation if you properly regularize the divergent series and apply analytic continuation.

I confess, I don't understand what you mean by "This "identity" as it stands here is simply wrong. But it can be useful for a calculation if you properly regularize the divergent series and apply analytic continuation." How can it be simply wrong, and yet useful?

ReplyDeleteAlso, as far as I can tell, the the Wikipedia piece is using the zeta function as defined here http://en.wikipedia.org/wiki/Riemann_zeta_function. So zeta (-3)=1^3+2^3+3^3+ . . .

In an earlier line, they write sum[abs(n)^(3-s)], then take the limit as s goes to zero, leaving sum[abs(n)^(3)] which looks like the usual zeta function with the argument -3.

So I still believe I'm right in saying that the Wikipedia includes the equation 1^3+2^3+3^3 +4^3+ . . . = 1/120. It may or may not be a correct equation, but it's what the page says.

Concerning the Riemann zeta function, I already answered this in my reply to Anonymous from January 24 at 3:02 AM.

ReplyDeleteIn short: The Riemann zeta function zeta(s) is an analytic function defined for s in the whole complex except s=1 where zeta(s) has a pole. In my reply from January 24 at 3:02 AM above, I linked the PDF file of a book where you can find various definitions of zeta(s) which are also valid for negative s. However, the series representation zeta(s) = 1^(-s) + 2^(-s) + 3^(-s) + ... is only valid if the real part of s is larger than 1 (so for s>1 on the real axis). That means the series can only be used to define zeta(s) for Re(s)>1, but other definitions are available for Re(s)<1 which may be derived from the series representation via analytic continuation.

The Wikipedia page on the Riemann zeta function does not state zeta(-3) = 1^3 + 2^3 + 3^3 + ... Instead, it says in the introduction: the "Riemann zeta function [...] is a function of a complex variable s that analytically continues the sum of the infinite series [...], which converges when the real part of s is greater than 1." That is exactly what I explained above. And also the "Definition" section of the Wikipedia article restricts the given series definition to Re(s)>1, stating that the "infinite series defines zeta(s) in this case". Nowhere in the Wikipedia page do you find "zeta(-3) = 1^3 + 2^3 + 3^3 + ..." or "zeta(-1) = 1 + 2 + 3 + ...", which would be wrong.

The other Wikipedia page on the Casimir effect uses the Riemann zeta function in order to sum 1^(3-s) + 2^(3-s) + 3^(3-s) + ... = zeta(s-3), which is valid for real parts of s larger than 4. This zeta(s-3) is then identified with the analytic continuation of the zeta function which is valid also for negative values of s (in contrast to the series!). So, while intermediary steps of the calculation where valid only for Re(s)>4, the limit for s->0 of the final result is taken in the end. Such a procedure is often applied when using regularization, and it is justified by knowing that the calculated quantity is an analytic function of s.

If I find more time later, I will comment on what I meant by an identity being "simply wrong and yet useful".

So, what did I mean by saying before "This 'identity' as it stands here is simply wrong. But it can be useful for a calculation if you properly regularize the divergent series and apply analytic continuation."?

ReplyDeleteThe "identity" 1 + 2 + 3 + ... = -1/12 is wrong, because the summation of a divergent series does not have a value and cannot be equal to a finite number.

However, the following statement is correct:

"The Riemann-zeta-regularized summation of 1 + 2 + 3 + ... yields -1/12."

If you mean a specific regularization for your divergent series, then you have to name it. Simply writing "1 + 2 + 3 + ..." implies standard summation by adding consecutively more and more terms, and this does not yield a finite value here.

What "Riemann-zeta-regularized" means is that you replace the original divergent series by the new series 1^(-s) + 2^(-s) + 3^(-s) + ... which converges for s>1. (I am now omitting "real part" for simplicity.) This new series is summed, it yields zeta(s). Then this function zeta(s) is analytically continued to negative values of s, where the series itself has no meaningful value any more. The result of these steps is zeta(-1) = -1/12. Mark: zeta(-1) is not equal to the original divergent series, but it follows from it through the regularization steps described here.

For a physics calculation, it is only safe to apply such Riemann-zeta regularization if the regulator s is introduced at a point in the calculation where everything is still mathematically well-defined, i.e. before arriving at a divergent series. Otherwise there are caveats such that the calculation might lead to right or wrong results, you never know in general, and manipulating divergent expressions is a dangerous thing to do.

By the way, look at the Wikipedia page for "1 + 2 + 3 + 4 + ⋯" (http://en.wikipedia.org/wiki/1_%2B_2_%2B_3_%2B_4_%2B_%E2%8B%AF). This page says that the series "can be manipulated to yield a number of mathematically interesting results", that it is "'summed' by zeta function regularization" ('summed' even in quotation marks), or that its "Ramanujan summation [...] is also -1/12". These formulations are different from stating plainly "1 + 2 + 3 + ... = -1/12", which is clearly avoided in the Wikipedia article.

The twelfth root of 2 is how we divide the musical scale, each note is 1/12th of the octave, wonder if there is a connection. Its often assumed that all music strictly cultural but the evidence does not support it, for example tone/tempo of voice carries emotional information you can understand even if you don't know the language and have no other clues, play a film in a foreign language while blindfolded and you can still tell what the characters are feeling by tone/tempo of voice alone. The variances in tempo and pitch that convey this information are indistinguishable from musical variances, we speak with music only some of which carries abstract associative meaning but most of which carries universally understood emotional state information. This suggests that the emotional content of music is conveyed by adherence to universal rules that are just as solid as the rules of physics with the 12th root of two the simplest generator of the frequencies for both tempo and tone.

ReplyDeleteAnalytical continuation is not a way to make divergent series converge. It just isn't.

ReplyDeleteAnalytical functions are by definition those that admit a power

expansion at every point. That means that *locally* you can write them

as the sum of their Taylor expansion. (And no, not all functions are

analytic, even if they admit infinitely many derivatives: sometimes

you can write the Taylor expansion but that does not equal the

function you started with. But I digress.)

Being analytic does not mean that the function is the same as its

Taylor expansion at a point. The Taylor expansion, in particular, may

start diverging at points where the function is perfectly well

defined. Taylor expansions have a well-defined convergence radius in

the complex plane, and that radius may be infinite (as it is for the

case of e^z, say) but it is often finite.

So, if you start with a given Taylor series, there can be a function

that the Taylor series was the expansion of (at a given point), and

that function may be well-defined on a larger domain than the Taylor

series was. But that doesn't mean that you are making the original

Taylor series converge. It still diverges: it's just that it only

represents the function _within_ its radius of convergence.

If you want to play with an example, take f(z)=1/(z-1). f(z) is

perfectly well defined at any complex number z, as long as it's not 1.

Now take its Taylor expansion at z=0, meaning

f(0) + f'(0)z + 1/2f"(0)z^2 + 1/6 f'''(0)z^3 + ...

Its radius of convergence is 1, which means that the Taylor series

*diverges* for any |z|>1. The fact that f(z) is well defined for *all*

|z|>1 doesn't magically make the series converge there. It's the

*function* that is well defined there, not the Taylor series. If you

had started with the Taylor series, and then realized that it was the

Taylor series of 1/(z-1), then you could say that f(z) was an

analytical continuation of the Taylor series. But again, doing that

doesn't make the series any more convergent than it was before. This

is stuff that has been figured out once and for all 200 years ago.

About 1+2+3... Evelyn Lamb has already said everything that there is to say: "There is a meaningful way to associate the number -1/12 to the series 1+2+3+4…, but in my opinion, it is misleading to call it the sum of the series."

End of story.

http://blogs.scientificamerican.com/roots-of-unity/2014/01/20/is-the-sum-of-positive-integers-negative/

I think the point is that no one is saying they converge. It's the same as for the series 1-1+1-1+1-1+...

ReplyDeleteIt doesn't converge, but if you include all the terms to infinity (not the limit as you go to infinity, but the value when infinite terms are included), it equals 1/2.

If you approach it by thinking in terms of convergence, as you point out, there's nothing you can do with zeta(-1), zeta (-3), or 1-1+1-1+1 . . . And yet, using the values that Euler, Reimann, and Ramanujan calculated for those things give you meaningful answers to real, testable theories.

I can't see a way to rectify those two things - either the equations and the theories are true, in some sense anyway, or they're each wrong in ways that precisely (to 12 or more decimal places) correct the errors the other one produces. The second option seems much harder to believe than the first.

I am sorry Physics Buzz, but you are talking nonsense. The statement "include all the terms to infinity" is devoid of meaning. And as Evelyn Lamb explained, you are misrepresenting what Ramanujan & co. did. I don't think there is anything more I can say at this point other than suggesting that you read the sentence "There is a meaningful way to associate the number -1/12 to the series 1+2+3+4…, but in my opinion, it is misleading to call it the sum of the series" again and again. Perhaps it would have been advisable to do so before blogging about this on a website that's supposed to be educational.

ReplyDeleteRamanujan (and Euler) wrote

ReplyDelete1+2+3+4…= -1/12

Ramanujan's exact words are ". . . the sum of an infinite number of terms of the series

1+2+3+4…= -1/12 . . ."

I am only repeating what he said.

See for yourself here http://books.google.com/books?id=Of5G0r6DQiEC&pg=PA53&dq=gratified#v=onepage&q&f=false

Are you saying he is wrong or that "=" means something other than equals?

Keep in mind, I'm saying

ReplyDeleteIf QED is true

and it relies on 1+2+3+4…= -1/12

then 1+2+3+4…= -1/12 must be true.

I have no reason to want 1+2+3+4…= -1/12 to be true. I am happy with it being infinite. But then what's going on with QED? I don't accept that luck can lead to such precise predictions.

Thanks, Davide, for your comments. I really like Evelyn Lamb's statement which you cited: You may assign -1/12 to the divergent series, but it is misleading to call this the sum of the series.

ReplyDeleteAnd, Buzz (are you the same as "Buzz Skyline" or a different person?), it is sad when the only remaining argument for a mathematical equation like 1+2+3+4+… = -1/12, is the correctness of theories in physics (like QED). As a physicist I tell you, there is no meaningful theory in physics which relies on "identities" like 1+2+3+4+… = -1/12. Whenever infinities and divergences are encountered in physics, they are properly regularized (and eventually renormalized, especially in QED). And that is also what Euler, Ramanujan & co. did: They attached a meaning to divergent series by choosing a certain regularization.

By saying that the theories rely on the equation, I'm saying that, as I understand it, the theories would make different predictions if the "sum of the infinite series" (as Ramanujan says) equaled something other than -1/12. If you can do without them, they why to physicists who do QED calculations tell me they need them?

DeleteI have also done a lot of QED calculations, and I have never needed to sum divergent series without a proper regularization. Probably the physicists you are speaking about just wrote down such "identities" with divergent series as a very simple picture of what they did, omitting all notion of regularization for sake of simplicity? Maybe they even wanted to impress you by showing you what kind of "crazy things" they are doing? (Things which aren't crazy at all, once you properly define your regularization procedure.)

DeleteAs I told you before, physicists tend to be pragmatic. So they might even work with "identities" like 1+2+3+4+... = -1/12 because they know that, in principle, it can also be done properly and mathematically. Anyway, theories in physics do not depend on such "identities" being true, they just depend on being regularizable and renormalizable such that their predictions are finite in the end.

Well, finite, and *correct* I assume, is what your after. I'm sure there are many ways to deal with infinities. I imagine the ways that give you precise answers are fewer.

DeleteSo your proposal is that the answer is more along the lines of a conspiracy of physicists who are telling us this stuff to impress. It's possible, I suppose, but hard to believe.

No conspiracy. But some people tend to oversimplify things when they present them to non-experts. Then they are presenting them in a way which is actually not correct, but behind the wrong picture is a true meaning which though would have been more complicated to explain in a correct way. That's what the Numberphile people have done in my opinion. I always try to avoid this, but sometimes it is hard.

DeleteThis is proof of my Theory of Dark Numbers. This theory postulates that Dark Numbers exist (but yet unobserved) to tame the large expansion of currently observable whole numbers 1+2+3+4+... so that the infinite series is equal to -1/12 instead of infinity. Tesla alluded to using Dark Numbers when he built his transport ray machine. According to my theory, the number line contains 4.9% ordinary numbers, 26.8% dark Robinson hyper-reals and 68.3% dark numbers.

ReplyDeleteBuzz, let me show you one way to better understand, at least in a qualitative way, how such divergent sums of positive numbers can be related to finite and sometimes even negative numbers.

ReplyDeleteLook at the physics case of the Casimir effect (http://en.wikipedia.org/wiki/Casimir_force#Derivation_of_Casimir_effect_assuming_zeta-regularization): Even when properly regularized by using the parameter s, how can a sum (and integral) of energy contributions, which are each individually positive, end up in a negative total Casimir energy at the end?

In order to explain this, let me first approximate the series representation of the zeta function by an integral which is easier to handle. Using the Euler-Maclaurin-Formula (http://en.wikipedia.org/wiki/Euler%E2%80%93Maclaurin_formula) in its 0th-order approximation, we can approximate

1^(-s) + 2^(-s) + 3^(-s) + 4^(-s) + ...

= 1/2 + integral from 1 to infinity of dx * x^(-s) + small remainder,

which is valid, as the series on the left-hand side, for s>1.

Now, as the integrand x^(-s) is positive all over the integration domain from 1 to infinity, one would expect this integral to be positive as well. And in the convergent case s>1, this is as expected. Using the primitive of the integrand, -1/(s-1) * x^(1-s), one gets:

integral = 1/(s-1) * ( 1 - infinity^(1-s) )

For s>1, the contribution from the upper boundary is infinity raised to a negative power, i.e. zero, and we and up with:

integral = 1/(s-1) > 0 for s>1.

But what happens in the divergent case, when s<1? Then the contribution from the upper boundary is divergent: infinity raised to a positive power. So the integral should be positive and infinite, as one would also expect from summing all natural numbers (the above series with s = -1) or all cubes of the natural numbers (series with s = -3). However, we are using a kind of regularization here which is based on analytic regularization. As the result 1/(s-1) for our integral is valid all over the complex half-plane where the real part of s is larger than 1, we analytically continue this result to all values of s (except s=1). Practically this means that the contribution from the upper boundary of the integral, 1/(s-1) * infinity^(1-s), is omitted although it diverges.

So our analytically regularized integral is still given by minus the primitive of the integrand at the lower boundery, which is now a negative contribution:

integral = 1/(s-1) = -1/(1-s) < 0 for s<1.

Summarizing, the negative and finite result is obtained by dropping the positive and infinite contribution from the upper boundary of the integral. This prescription to drop such an infinite contribution arises from the analytic continuation in the parameter s.

In physics (example of the Casimir effect) such a prescription makes sense because constant energy shifts (even if they are infinite) are irrelevant for the Casimir force.

Bernd, my boss happened to stop by and say basically the same thing (he's a physicist who has a lot of experience with quantum field theory) and illustrate it using the Casimir Effect. I have to admit, I'm coming around to see what you're saying. It's easier to understand with a white board handy. In any case, I'll have to think about it for a while before I have anything much to say (or ask).

ReplyDeleteAlong the way, I happened to find a relatively simple way to evaluate zeta functions that I'd never seen before, for odd, negative values of n. I guess it doesn't matter, really, but it happens to make calculating some Bernoulli numbers (B_n, for n even) relatively quick and easy. I have no idea whether that's useful, interesting, or new. It entertained me though.

Wondrful exchange guys. Thanks for the knowledge you've been sharing with us, and for the effort that takes.

ReplyDeleteDefinition error. The sum of n to infinity diverges, goes really big. The "regularization" converges to -1/12. The technique is proven, duh, it would be odd if they were using the regularized sum without a proof. Its like saying 2+2=10. Very misleading unless you realize that the equation has been manipulated by being turned into 2, base ten, +2, base ten, equals 10 base 2. Although the regularization is legitimate, you can't say the sum adds to -1/12. You have to say the sum can be regularized through the Riemann zeta function and through a variety of other methods to equal -1/12. Although for all intents and purposes no ones checking the addend up to infinity to make sure that it equals what we say it does so feel free to say it equals -1/12, but make sure add in that regularized part if your talking with any self respecting mathematician or physicist.

ReplyDeleteFor me, not a mathematician, this is a perfect example for that you can prove anything with infinity mathematics.

ReplyDeleteIn my view, infinity does not exist (in the real world), nor does 0.

Infinity = anything / 0

so

0 = anything / infinity.

Physical argument: the smallest thing (measure) is the planck dimension (planck lenght, planck time etc.) so there cannot be an infinite number of since the beginning of time (the big bang).

I'm afraid you're greatly confused.

ReplyDelete1. Just because Euler derived something doesn't make it true. Euler was known for "breaking the rules" all too often - that was surely part of his genial nature - but it doesn't mean that all of his results and proofs are correct, in fact his handling of infinite diverging sums is not strictly mathematically correct.

Also, n appeal to authority is a fallacy.

2. It's true that 1+2+3... results in the so-called Ramanujan sum of -1/12. But a Ramanujan sum is not at all the same as *a* sum in a traditional sense. Look it up.

3. You're engaging in trickery by juggling different definition of sums.

If we're talking about traditional sums, then no: divergent infinite series cannot be summed up and -1/12 is certainly not a sum of any such series.

Now, we can extend the definition of "summation" to cover the assignment of finite numbers to infinite divergent series. But in such a case it *must* be clear that we're talking about something entirely different than traditional summation. Because 1+2+3...=-1/12 is only mindblowing when we're thinking of traditional sums (in that case it also happens to be incorrect).

4. Now, certainly it must be admitted that the persistence with which -1/12 (and other numbers) pops up when all those different kinds of operations are performed (legally or not), as well as the fact that it is applicable to the real world does mean that it's not merely a fluke and that there is indeed a deep connection between these things on a fundamental level.

But this doesn't justify the bad math, sorry.

PS: please read the 3rd paragraph at books.google.de/books?id=n8Mmbjtco78C&pg=PA73

DeleteI totally agree. And I refer to my previous comments above, where I have made similar statements.

DeletePlease note that the discussion about this topic actually went on in two new blog posts by the same author:

http://physicsbuzz.physicscentral.com/2014/01/redux-does-1234-112-absolutely-not.html

[titled "Correction: Does 1+2+3+4+ . . . =-1/12? Absolutely Not! (I think)"]

and

http://physicsbuzz.physicscentral.com/2014/02/so-what-does-1234-equal-we-give-you_11.html

[titled "If Not -1/12, What Does 1+2+3+4+... Equal? We Give You the Answer "]

You may see from these blog posts (and their comments) that the author changed his mind several times about this topic. Unfortunately, he did not link his follow-up posts from this first one.

Thanks, Dr. Jantzen!

DeleteProblems of summation divergent series is solved! https://m4t3m4t1k4.wordpress.com/2015/02/14/general-method-for-summing-divergent-series-determination-of-limits-of-divergent-sequences-and-functions-in-singular-points-v2/

ReplyDeletehttps://m4t3m4t1k4.wordpress.com/2015/11/25/general-method-for-summing-divergent-series-using-mathematica-and-a-comparison-to-other-summation-methods/

ReplyDeletePlease please can someone point out the flaw in my reasoning. This theory is completely WRONG.

ReplyDeleteIf I take the series S = 1 + 2 + 3 + 4 + ....

and the series 4S = 4 + 8 + 12 + 16 + ...

And subtract the 2, I cannot just simply shift and then pretend everything works out ok. Why shift the bottom row with intervals that line up numerically with the top row? Why not just shift it all an infinite amount and end up with:

S - 4S = (1 + 2 + 3 + 4 + 5 + 6) - 4?

The amount of shift is completely arbitrary and is just done for convenience sake. It completely disregards the rate of numerical increase.

We all know that a function f(x) = 2x [x limit to infinity] is larger than the function f(x) = x [x limit to infinity]. It's absolutely important that the numbers in each position are subtracted with the corresponding position.

Please tell me why I am wrong, or tell me how some string theory is actually based off of this??