The infinite sum which equals -1/12

S. Ramanujan and G. H. Hardy

On 16 January, 1913, Srinivasa Ramanujan, a clerk living in Madras, India, sent a letter to Oxford professor Godfrey Harold Hardy, which contained many of his intuitive mathematical musings, one of which led to the conclusion:

\displaymath \sum_{n=1}^{\infty} n = 1 + 2 + 3 + 4 + 5 + ... = \frac{-1}{12}

In normal mathematics, this bizarre result would be just that — a bizarre result. In high school math, as well as in most university math courses, a response of \frac{-1}{12} would get a mark of zero. On the face of it, the series is clearly divergent, and there is no reason to doubt that the actual answer should be \infty.

What is also clear is that Ramanujan and Hardy were not dummies, having helped to revolutionize early 20th century mathematics in England, as well as making people begin to take notice that India has, for a long time, made significant advancements in mathematics. It is also important to note that this bizarre result is used today in string theory, in the context of the Riemann Zeta function.

\displaystyle \zeta(z) = \sum_{n=1}^\infty \frac{1}{n^z} = \frac{1}{1^z}+\frac{1}{2^z}+\frac{1}{3^z}+...

The difference is that the Zeta function works with complex numbers. The zeta function does not replace ordinary summation. Also, a finite sum still exists in the real numbers if their partial sums are shown to converge.

The zeta function comes from a well-known summation which goes something like:

\displaystyle \frac{1}{1-z} = 1 + z + z^2 + z^3 + z^4 + ... ; -1 < z < 1

The restriction on z is important if the series is to converge. When z wanders outside those boundaries, the series diverges, and the sytem breaks down, and we can’t find the sum. The zeta function, however, attempts to generalize this equation for any value of z.

The value of these sums have meaning in a particular mathematical context. It is as if you encountered the imaginary number i = \sqrt{-1}, and said that the result is meaningless, since square roots are supposed to only return positive numbers. But in math involving the complex plane, it is quite meaningful, and enjoys actual use in the design of integrated circuits, for instance.

To see how Ramanujan encountered -1/12 as the sum of 1 + 2 + 3 + …, you can do normal summation, pretending that the series we are about to encounter always converge (and none of the following do):

\displaystyle \frac{1}{1 + 1} = 1 - 1 + 1 - 1 + 1 - 1 + ... = \frac{1}{2}

As you calculate partial sums, you get 1, 0, 1, 0, 1, …, and this never settles on any single value. This series is divergent. The 1/2 “sum” can be seen as an average of 0 and 1, and this actually is called a Cesaro sum.

Now suppose we squared this:

\displaystyle \left(\frac{1}{1 + 1}\right)^2 = (1 - 1 + 1 - 1 + 1 - 1 + ...)^2 = 1 - 2 + 3 - 4 + ... = \frac{1}{4}

The sum 1 – 2 + 3 – 4 … is the product of  the square of (1 - 1 + 1 - 1 + 1 ...)^2, which can be seen by using a grid to help you square them:

× 1 -1 1 -1 1
1 1 -1 1 -1 1
-1 -1 1 -1 1 1
1 1 -1 1 -1 1
-1 -1 1 -1 1 1
1 1 -1 1 -1 1

Then, you may add numbers from that table belonging to the same diagonal. Above, they happen to be the same colour. The 1’s and -1’s in gray are incomplete diagonals due to the limitations of a finite table.

You get: 1 – 2 + 3 – 4 + 5 – … = \displaystyle \left(\frac{1}{2}\right)^2 = \frac{1}{4}, which we will refer to as S_2.

Now, let’s get back to the original “sum”: S_1 = 1 + 2 + 3 + 4 + 5 + ... = -1/12. We can re-generate S_2 using only S_1 as follows:

\begin{aligned} S_1   & = & 1 + & 2 + & 3 + & 4 + & 5 + ... \\ 4S_1 & = &    &    4  +&        & 8  + &  ... \\ S_1 - 4S_1 & = & 1 - & 2 + & 3 -  &  4 + & 5 - ... \end{aligned}

… and finally, using our finding that S2 = 1/4, we arrive at the original incredible sum:

\begin{aligned} -3S_1 & = & \frac{1}{4} \\ S_1 & = & \frac{-1}{12} \end{aligned}

The sum is verified by the Riemann Zeta function for \zeta(-1).

The only story I am aware of where giving away the “answer” doesn’t matter.

The feeling I get when I see a result like this is like when reading the Douglas Adams book Hitchhiker’s Guide To The Galaxy, and then being told by their whizzbang computer that the answer to life, the universe and everything was 42. You feel no more enlightened than before you asked the question. But then that is because the answer is not understood. And for that matter, even the question may need some work.

Mathematicians such as Euler and Ramanujan had endeavoured to unlock the mysteries of these strange results.

We know that \zeta(z) works best for z > 1. For all z < 1 (especially negative numbers), \zeta(z) does not return a direct value, but instead the mathematician needs to base a conjecture of what the value could be, based on analytical continuation. It turns out, this is far from guesswork since, for any negative z in the set of complex numbers, there is always exactly one answer that analytic continuation ever offers. \zeta(-1) = -1/12 is the result when z = -1.

The zeta function also equals zero in strange places, such as when z is a negative even integer. The reader is reminded that the set of complex numbers is utterly vast, subsuming the entire set of real numbers as a subset of the complex numbers. This is because a complex number C is defined as C = a + bi, where a and b belong to the set of real numbers, and i, once again, is \sqrt{-1}. If you allow b = 0, then C is a real number.

According to Riemann’s Hypothesis, the other set of zeroes for the zeta function lie entirely in the set of complex numbers where C = \frac{1}{2} + bi, for some value of b. In other words, all other zeroes for the Riemann zeta function are of the form \zeta\left(\frac{1}{2} + bi\right) for some real number b.

Now, this is a hypothesis, which hasn’t been proved wrong yet, but it also hasn’t been proved to be correct generally. The Clay Mathematics Institute has made this one of their millenium prize problems, offering one million dollars to anyone who can solve it. If this is proven, other connected conjectures are proven along the way, such as the Goldbach Conjecture, and the Twin Prime Conjecture.

The locations of the zeroes of the zeta function appear to bear some relationship to the number of primes in an arithmetic progression with a fixed difference k. Proving Riemann will also validate the zeta function as useful for studying prime numbers, as had been done by Leonard Euler.

 

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