Warning !Genius !!!

Srinivasa Ramanujan
(1887 - 1920)

By denoting (x)_n the value : (it's Pochhammer's symbol), we get :















Phew!

Slices of his life

With Ramanujan, we reach the quintessence of the study of Pi. He is the master of all research in the 20th century in this domain, let's not be afraid to acclaim it!
I could spend days talking about this unrecognise genius who nevertheless left a magistal work still yet not understood in a few domains. His life is furthermore a true novel....

But let us start by the beginning :

Srinivasa Ramanujan was born the 22 decembre 1887 in the town of Erode in the south of India in a poor family. His dad was accountant. His mathematic precocity was quickly noticed and at seven, he recieved a bursary for the secondary school of Kumbakonam (!). It is said that he would recite mathematical formulae to his classmates and that he knew a surprising amount of the decimals of Pi!
Since the start of his study on trigonometry, he discovered cos and sin, found all the relation linking the two and showed to be very dissapointed when he learned that they were already known!!!
When he was 12, Ramanujan master a huge and dense book: Plane Trigonometry by Loney. When he was 15, he got hold of Synopsis of elementary results in pure and applied mathematics by G.S.Carr, a list of 6165 theorems states often without proofs. We assume that it's from this book inspired him abd his bad habit of not giving out proofs with his results!
In fact at that period he was so obssesed with his research that he failed all his exams!
Luckly, after his wedding in 1909, he received a monthly sum from a rich patron of the art passionate with mathematics (R. Rao) on the recomodation of the indians mathematicians who appriciate his discovery already written in what we commonly call his notebooks.
Gaining a stable job in 1912 as a beaurocrat at Madras' counter, he was encouraged by his managers to send his result to 3 distinguished british mathematicians, of which only G. Hardy replied to his letter dated 16/01/1913.
In fact, when Hardy and his collegue Littlewood examined a few of the 120 formulae and theorems send by Ramanujan, their conviction was made a few hours later: they were looking at a genuis! (Hardy had build a "scale of pure capacities" on which he situated himself at 25, gave 30 to Littlewood, and 80 to Hilbert, shinning figure of the german mathematics from the start of the century. Ramanujan was immediatly estimated at 100 !!!).
Hardy described later that the intellectual discovery of Ramanujan and its consequences as the only "romantic" evenment of his life....
When he looked at Ramanujan's formulae, he was disconcert and had no idea on how to proof the. But, he maintained, "they must be ture because he they weree not, nobody in the world would have enough imagination to invent them!"m
he made Ramanujan come to England and work with him for the next five fruitful following years on the properties of several arithmetics functions. Srinivasa became the first Indian to be member of the Royal Society in 1918 and of Trinity College (Cambridge).
Unfortunatly, and its such a shame, Ramanujan was strictly vegetarian (due to a promise made to his mother!), and in England in the full blow of war, his needs were hard to fufill... After the war, in 1919, he came back in India seriously ill from a tuberculose and a lack of vitamines (We all know how wet Brittain is!).
His work would stay of great quality despite his suffering, but he drew his last breath the 26 april 1920, he was only 32.

A little anecdote very well known by Hardy :

I remember once going to see him when he was ill at Putney. I had ridden in a taxi cab number 1729 and remarked that the number seemed to me rather a dull one, and that I hoped it was not an unfavorable omen.
"No" he replied "it is a very interesting number; it is the smallest number expressible as the sum of two cubes in two different ways." !!
I then asked him if he knew what was the next one. He thought about it for a minute and told me that he could not see any close one... In fact , the next one is several thousands later!! (4104)

Around

Ramanujan was passoniated with pi. Many of his results involve our favourite constant...
Ramanujan wrote down his work in his notebooks as I stated above. Unfortunatly many formulae are written in with non-standards notation and without any proofs. For more than 80 years, several mathematicians (Bruce Berndt for the moment) tried to decipher his coded notebooks for the happiness of Science!
Because Ramanujan worked on the modular equations. But what exactly are they?
I will 'borrow' a very clear example from the book Les mathématiciens to clarify the definition.

A modular equation is an equation that satisfy a modular function (q) where the variable q intervine with various power, for example (qⁿ). n denotes here the order of the modular equation.
Consider for example the modular equation of 7th ordre (n=7) :

We then look for the solution to this equation. For our case, we have:

(don't ask me how it was found...)
Up do here, no hints of Pi...
Well, we call for singular values from the values of the modular function (q) which satisfy some extra properties. For example, if we define for p integer:

Straight away we see that the bigger p is, the smaller the exponential and the product in (q) tends toward 16q
So, if we take , we get a number that coincide on the first decimals with Pi !
Of course, the number of decimals increase with p. We can see the whole advantage in having a relation between (q) and (q ^p), this last number being closer to Pi than the first (because the exponential is even smaller!)
The amazing thing about this theory, is that the singular value does not depend at all on Pi even with their definition
Ramanujan was a great specialist of those values and calculated them in a remarquable way. In his lettler to Hardy, he said:

which allows us to get 20 decimals of Pi.
K₂₄₀ allowed his to get the first million decimals of Pi !!
But he never did any research on the algorithm that we could build from them. The Borwein brothers took up that idea, here is how they proceded.

Principle of the proof of the Borweinein

If we look at the Borwein's proof for the algorithm of second order, we can see cleary that we consider the modular equation of second degree:

a²+b²=c²

with a(q)=₃⁴(q)+₂⁴(q), b(q)=₄⁴(q), c(q)=2₂²(q)₃²(q) and the Thêta Functions :



This equation lead to let and to find the true modular equation of second degree useful here :

whose solution is well known according to the definition of s(q) (in fact we started here from the solution to get to the modular equation.)
The initial value s₁ correspond to a first singular value, the sequence s_n hence allows to calculate a sequence of singular values.
The rest now like above consist of finding a way to come back to Pi, since logarithm did the job in the first example. The role here is played by the _p(p²r), and then we find from it an algorithm. the justification is given by the apparition of the function in Legendre's equation.
In all, a lovely little theory....

While we're talking about Ramanujan's formulae :

This page is a little bit special as it does not contain any proofs. I just reproduces the intermediate steps of the detailled calculation on the Borwein's canadians page. a part form the general formula of the formation of series of type Ramanujan found by the Borwein and written on my page dedicated to them, I prefer allowing you to see their site....

Other results by Ramanujan :

So, what else ?
There's first of all numberous approximation of Pi that he found in a prodigious way! They are written in the page on approximations of Pi as well as the famous result showing that is close to an integer...
Otherwise, there doesn't unfotunatly exist much information on the results of Ramanujan. Nevertheless, here is what I could find:

Prime numbers (function Tau of Ramanujan) :

Be so that , then for GCD(n,n')=1,
This first property allows to calculate the function Tau for all the product of prime numbers knowing the value Tau(p).
And since the naturals can be expressed as the product of prime numbers, the function Tau is completly evaluated on N if we can evaluate it for the powers of prime numbers, which is what the following theorem does by recurrence:

This function Tau has quite a few more properties:

There exists some congruence relations for 7, 23 and 691. For instance, .

Moreover, Ramanujan conjectured that for p prime.

Note that the relation of congruence is only true if k is without quadratic remainders modulo 23, that is that there exists no integers x such that x²=k[23]. Since there are (p-1)/2 quadratic remainders for p>2, we can deduce that on average, one natural integer N in two is such that tau(N) is divisible by 23.


Pierre Deligne (belge and not french) showed in 1971 that the conjecture above was one of the consequence of the conjectures by Weil. And since he obviously didn't want to leave it at that, he proved them in 1973, which was responsible in part for his Fields Medal in 1978.

And we also have :

where is the sum of eleventh power of the dividers of n.
For more information on sigma, cf.A013959

Landau-Ramanujan's constant :



The proof is self obvious if you do a variable change t=x-1, then write the finite expansion of Ln(1-t), and by justifying the change between sum and integral. I did not look for a₂,but it seems completle doable!

Spread of prime numbers :

where c is the root of the function Li defined by and is the Möbius function defined by =1 if k=1, (-1)^r of k is the product of r distinct prime numbers, 0 otherwise.

For more information on the function pi(x), cf.A000720 on Möbius' function, cf.A008683 and the relation between Möbius and Pi, cf arithmetic functions page.

Equations on the theory of numbers :


Ramanujan gave himself the problem to find the solutions to the equation :

2^N-7=X²

The computers have been able to find up till N=10⁴⁰ but only the solutions (N=3,4,5,7,15) were confirmed. We have in fact proved recently that they are the only valid !

For more information on this equation, cf.A060728

An other problem by Ramanujan ? But of course :
Find for example all the solutions of n!+1=x²

(I unfortunatly don't have the list...)