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Infinity or -1/12?

Recently a very strange result has been making the rounds. It says that when you add up all the natural numbers

1+2+3+4+...

then the answer to this sum is -1/12. The idea featured in a Numberphile video (see below), which claims to prove the result and also says that it's used all over the place in physics. People found the idea so astounding that it even made it into the New York Times. So what does this all mean?

The maths

First of all, the infinite sum of all the natural number is not equal to -1/12. You can easily convince yourself of this by tapping into your calculator the partial sums

$S_1 = 1$

$S_2 = 1+2 = 3$

$S_3 = 1+2+3=6$

$S_4 = 1+2+3+4=10$

$...$

$S_ n = 1+2+3+4+ ... +n,$

and so on. The $S_ n$ get larger and larger the larger $n$ gets, that is, the more natural numbers you include. In fact, you can make $S_ n$ as large as you like by choosing $n$ large enough. For example, for $n=1000$ you get

  \[ S_ n = 500,500, \]    

and for $n = 100,000$ you get

  \[ S_ n = 5,000,050,000. \]    

This is why mathematicians say that the sum

  \[ 1+2+3+4+ ...  \]    

diverges to infinity. Or, to put it more loosely, that the sum is equal to infinity.

Srinivasa Ramanujan

Srinivasa Ramanujan

So where does the -1/12 come from? The wrong result actually appeared in the work of the famous Indian mathematician Srinivasa Ramanujan in 1913 (see this article for more information). But Ramanujan knew what he was doing and had a reason for writing it down. He had been working on what is called the Euler zeta function. To understand what that is, first consider the infinite sum

$S = 1+1/4+1/9+1/16+ ... .$

You might recognise this as the sum you get when you take each natural number, square it, and then take the reciprocal:

$S = 1+1/2^2+1/3^2+1/4^2... .$

Now this sum does not diverge. If you take the sequence of partial sums as we did above,

$S_1 = 1$

$S_2 = 1+1/2^2 = 5/4=1.25$

$S_3 = 1+1/2^2+1/3^2 = 49/36 = 1.361...$

$...$

$S_ n =1+1/2^2+1/3^2 + ... + 1/n^2,$

then the results you get get arbitrarily close, without ever exceeding, the number $\pi ^2/6 = 1.644934... .$ Mathematicians say the sum converges to $\pi ^2/6$, or more loosely, that it equals $\pi ^2/6.$

Now what happens when instead of raising those natural numbers in the denominator to the power of 2, you raise it to some other power $x$? It turns out that the corresponding sum

  \[ S(x) = 1+1/2^ x+1/3^ x+1/4^ x...  \]    

converges to a finite value as long as the power $x$ is a number greater than $1$. For every $x > 1$, the expression $S(x)$ has a well-defined, finite value. $S(x)$ is what’s called a function, and it’s called the Euler zeta function after the prolific 17th century mathematician Leonhard Euler.

So far, so good. But what happens when you plug in a value of $x$ that is less than 1? For example, what if you plug in $x=-1$? Let’s see.

  \[  S(-1) = 1+1/2^{-1}+1/3^{-1}+1/4^{-1}...  \]    
  \[  = 1+2+3+4+ ... . \]    

So you recover our original sum, which, as we know, diverges. The same is true for any other values of $x$ less than or equal to 1: the sum diverges.

Extending the Euler zeta function

As it stands the Euler zeta function S(x) is defined for real numbers x that are greater than 1. The real numbers are part of a larger family of numbers called the complex numbers. And while the real numbers correspond to all the points along an infinitely long line, the complex numbers correspond to all the points on a plane, containing the real number line. That plane is called the complex plane. Just as you can define functions that take real numbers as input you can define functions that take complex numbers as input.

One amazing thing about functions of complex numbers is that if you know the function sufficiently well for some set of inputs, then (up to some technical details) you can know the value of the function everywhere else on the complex plane. This method of extending the definition of a function is known as analytic continuation. The Euler zeta function is defined for real numbers greater than 1. Since real numbers are also complex numbers, we can regard it as a complex function and then apply analytic continuation to get a new function, defined on the whole plane but agreeing with the Euler zeta function for real numbers greater than 1. That's the Riemann zeta function.

But there is also another thing you can do. Using some high-powered mathematics (known as complex analysis, see the box) there is a way of extending the definition of the Euler zeta function to numbers $x$ less than or equal to 1 in a way that gives you finite values. In other words, there is a way of defining a new function, call it $\zeta (x),$ so that for $x>1$

$\zeta (x) = S(x) = 1+1/2^ x+1/3^ x+1/4^ x...,$

and for $x\leq 1$ the function $\zeta (x)$ has well-defined, finite values. This method of extension is called analytic continuation and the new function you get is called the Riemann zeta function, after the 18th cenury mathematician Bernhard Riemann. (Making this new function give you finite values for $x \leq -1$ involves cleverly subtracting another divergent sum, so that the infinity from the first divergent sum minus the infinity from the second divergent sum gives you something finite.)

OK. So now we have a function $\zeta (s)$ that agrees with Euler’s zeta function $S(x)$ when you plug in values $x>1$. When you plug in values $x \leq 1$, the zeta function gives you a finite output. What value do you get when you plug $x=-1$ into the zeta function? You’ve guessed it:

  \[ \zeta (-1)=-1/12. \]    

If you now make the mistake of believing that $\zeta (x) = S(x)$ for $x=-1$, then you get the (wrong) expression

  \[ S(-1) = 1+2+3+4+ ... = \zeta (-1) = -1/12. \]    

This is one way of making sense of Ramanujan’s mysterious expression.

The trick

So how did the people in the Numberphile video "prove" that the natural numbers all add up to -1/12? The real answer is that they didn’t. Watching the video is like watching a magician and trying to spot them slipping the rabbit into the hat. Step one of the "proof" tries to persuade you of something rather silly, namely that the infinite sum

  \[ 1-1+1-1+1-.... \]    

is equal to $1/2.$

The video doesn’t dwell long on this and seems to imply it’s obvious. But let’s look at it a little closer to see if it makes sense at all. Suppose that the sum $1-1+1-1+1-1....$ has a finite value and call it $Z$. Adding $Z$ to itself you get the infinite sum

  \[ Z+Z = 1-1+1-1+1-....+1-1+1-1+1-... . \]    

But this is just the original sum, implying

  \[ Z+Z=2Z = Z. \]    

Since $Z=1/2,$ it follows that $1/2=1,$ which is nonsense. So the assertion that the infinite sum $1-1+1-1+1-....$ can be taken to equal to 1/2 is not correct. In fact, you can derive all sorts of results messing around with infinite sums that diverge. It’s a trick!

The physics

But how did this curious, wrong result make it into a physics textbook, as shown in the video? Here is where things really get interesting. Suppose you take two conducting metallic plates and arrange them in a vacuum so that they are parallel to each other. According to classical physics, there shouldn't be any net force acting between the two plates.

Casimir effect

Illustration of the Casimir effect. Image: Emok.

But classical physics doesn't reckon with the weird effects you see when you look at the world at very small scales. To do that, you need quantum physics, which tells us many very strange things. One of them is that the vacuum isn't empty, but seething with activity. So-called virtual particles pop in and out of existence all the time. This activity gives a so called zero point energy: the lowest energy something can have is never zero (see here for more detail).

When you try to calculate the total energy density between the two plates using the mathematics of quantum physics, you get the infinite sum

  \[ 1 + 8 + 27 + 64 +... . \]    

This infinite sum is also what you get when you plug the value $x=-3$ into the Euler zeta function:

$S(-3) = 1 + 1/2^{-3} + 1/3^{-3} + 1/4^{-3} + ... = 1+ 8 + 27 + 64 +... .$

That’s unfortunate, because the sum diverges (it does so even quicker than than $S(-1)$), which would imply an infinite energy density. That’s obviously nonsense. But what if you cheekily assume that the infinite sum equals the Riemann zeta function, rather than the Euler zeta function, evaluated at $x=-3$? Well, then you get a finite energy density. That means there should be an attractive force between the metallic plates, which also seems ludicrous, since classical physics suggests there should be no force.

But here’s the surprise. When physicists made the experiment they found that the force did exist — and it corresponded to an energy density exactly equal to $\zeta (-3)$!

This surprising physical result is known as the Casimir effect, after the Dutch physicist Hendrik Casimir.

Take a moment to take this in. Quantum physics says the energy density should be

  \[ S(-3) = 1+8+27+64+... . \]    

That’s nonsense, but experiments show that if you (wrongly) regard this sum as the zeta function $\zeta (x)$ evaluated at $x=-3$, you get the correct answer. So it seems that nature has followed the ideas we explained above. It extended the Euler zeta function to include values for $x$ that are less than 1, by cleverly subtracting infinity, and so came up with a finite value. That’s remarkable!

The reason why we see $\zeta (-1)$ and $S(-1)$ in the Numberphile video and the physics textbook, rather than $\zeta (-3)$ and $S(-3),$ is that when you imagine the Casimir effect as happening in one dimension (along a line rather than in 3D), the energy density you calculate is $\zeta (-1)$ rather than $\zeta (-3)$.

So why did the Numberphile people publicise this strange "result"? They certainly know about the analytic continuation that makes the function well-defined, but that was something that was a little too technical for their video. Knowing they had the analytic continuation method, that would make the final result OK, hidden in their back pocket, they went ahead with their sleight of hand. In doing so they got over a million hits and had the world talking about zeta functions and mathematics. For this they should be congratulated. The mathematics of zeta functions is fantastic and what we described here is just the start of a long list of amazing mathematical properties. In bringing mathematics and physics to the public we always have to make choices about what we leave out and what we explain. Where to draw that line is something we all have to leave to our consciences.


About the authors

David Berman is a Reader in Theoretical Physics at Queen Mary, University of London. He previously spent time at the universities of Manchester, Brussels, Durham, Utrecht, Groningen, Jerusalem and Cambridge as well as a year at CERN in Geneva. His interests outside of physics include football, music and theatre and the arts.

Marianne Freiberger is editor of Plus.

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