How to reach for the unreachable

What are you going to do with “knowledge” when all that is needed in the world, today, is entertainment?

I’ve been in love with Srinivasa Ramanujan and his work for lives together. The same way you care of a weak brother or a friend who cannot stand for himself. And today he is considerate a mathematical genius “alla pari” with Eurel or Jacobi. He wrote in his lost and found notebook a hundred pages of several mathematical formulas mostly proven later by the work of George Andrews and Bruce C. Berndt. In fact, he was accustomed to writing maths and formulas with neither motivation or proof while formal verification was superfluous. This is “zenish”, I might say. Zen is the direct experience, of our true nature expressing itself, moment by moment, and it is a tremendous disciplined practice, like Math.

Before Srinivasa Ramanujan came to England he lived on the charity of his friends back in India, and when he became ill, he had surgery thanks to the doctor who agreed to perform it free of charge. When he wrote to a famous English mathematician: “I am already a half-starving man,” he really meant it as he continued: “To preserve my brains I want food,” and was not talking about food for the soul.

After he left Madras, while in England experiencing great difficulties mostly related to his religion, Hinduism, which also requires a vegetarian diet, he attempted, in a fit of desperation, to throw himself in front of a subway train.

Srinivasa Ramanujan

Probably one of his rare miscalculation.

All his formulas were listed consecutively without proofs, apparently without an order or a meaning. To the public, one of his most remembered results is: the sum of all positive natural numbers is equal to minus one-twelfth. The result gives goosebumps because if we want to calculate the sum of the all the positive integers:

                   1 + 2 + 3 + 4 + … = ?

at first sight we could say that the Sum is quite a big number, being the set N = (1, 2, 3, 4, …} an infinite set.

Tho ∞ is a bit of abstraction it is in good company with all the rest of Math.

                   1 + 2 + 3 + 4 + … = ∞

Srinivasa Ramanujan, in a more general way than Euler, explained that

                   1 + 2 + 3 + 4 + … = -1/12

and that sounds crazy.

You can with F# try for your self:

let N = [1..10000]
let sum = List.sum N
printfn "%A" sum

Just for the first 10000 numbers the value is really high:

50005000

It is important to mention that the Ramanujan sums are not the sums of the series in the usual sense. In fact the result is mentioned as -1/12 ( ℝ).

Consider Euler’s zeta function S(x):

If we take x= 2

 S(x) = 1 + 1/2^x + 1/3^x + 1/4^x…

the results get arbitrarily close, without ever exceeding, the number pi²/6 = 1.644934…

let N = [1.0f..10000.0f]
let sum = List.sum (N  |> List.map (fun x -> 1.0f/x**2.0f))
printfn "%A" sum
1.64472532f

when you plug in values x<=1, it gives you a finite output.

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

What value do you get when you plug x=-1 into the zeta function?

S(-1) = 1+2+3+4+ ... =  -1/12

This a way of making sense of Ramanujan’s mysterious result. And mysterious was his death, back to India, while he was showing signs of chronic fever and disease.

Sengai last poetry would have been a closing statement on his life:

         To what shall I compare this life of ours?
         Even before I can say
         it is like a lightning flash or a dewdrop
         it is no more.